copyright 1996 lawrence c. marsh powerpoint slides for undergraduate econometrics by lawrence c....

PowerPoint Slidesfor

Undergraduate Econometricsby

Lawrence C. Marsh

To accompany: Undergraduate Econometricsby R. Carter Hill, William E. Griffiths and George G. Judge

Publisher: John Wiley & Sons, 1997

The Role of Econometrics

in Economic Analysis

Chapter 1

Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.

Using Information:

1. Information from economic theory.

2. Information from economic data.

The Role of Econometrics 1.2

Understanding Economic Relationships:

federalbudget

Dow-JonesStock Index

tradedeficit Federal Reserve

Discount Rate

capital gains taxrentcontrollaws

short termtreasury bills

power oflabor unions

crime rate

inflation

unemployment

money supply

economic theory

economic data } economicdecisions

To use information effectively:

*Econometrics* helps us combine economic theory and economic data .

Economic Decisions 1.4

Consumption, c, is some function of income, i :

c = f(i)

For applied econometric analysis this consumption function must be specified more precisely.

The Consumption Function 1.5

demand, qd, for an individual commodity:

qd = f( p, pc, ps, i )

supply, qs, of an individual commodity:

qs = f( p, pc, pf )

p = own price; pc = price of complements;ps = price of substitutes; i = income

p = own price; pc = price of competitive products;ps = price of substitutes; pf = price of factor inputs

demand

supply

Listing the variables in an economic relationship is not enough.

For effective policy we must know the amount of changeneeded for a policy instrument to bring about the desiredeffect:

How much ?How much ?

• By how much should the Federal Reserve raise interest rates to prevent inflation?

• By how much can the price of football tickets be increased and still fill the stadium?

Answering the How Much? question

Need to estimate parameters that are both:

1. unknown and

2. unobservable

Average or systematic behaviorover many individuals or many firms.

Not a single individual or single firm.

Economists are concerned with theunemployment rate and not whethera particular individual gets a job.

The Statistical Model 1.9

The Statistical Model

Actual vs. Predicted Consumption:

Actual = systematic part + random error

Systematic part provides prediction, f(i),but actual will miss by random error, e.

Consumption, c, is function, f, of income, i, with error, e:

c = f(i) + e

Need to define f(i) in some way.

To make consumption, c, a linear function of income, i :

f(i) = 1 + 2 i

The statistical model then becomes:

c = 1 + 2 i + e

The Consumption Function 1.11

• Dependent variable, y, is focus of study (predict or explain changes in dependent variable).

• Explanatory variables, X2 and X3, help us explain

observed changes in the dependent variable.

y = 1 + 2 X2 + 3 X3 + e

The Econometric Model 1.12

Statistical Models

Controlled (experimental) vs.

Uncontrolled (observational)

Uncontrolled experiment (econometrics) explaining consump-

tion, y : price, X2, and income, X3, vary at the same time.

Controlled experiment (“pure” science) explaining mass, y :

pressure, X2, held constant when varying temperature, X3,

and vice versa.

Econometric model

• economic modeleconomic variables and parameters.

• statistical modelsampling process with its parameters.

• dataobserved values of the variables.

• Uncertainty regarding an outcome.• Relationships suggested by economic theory.• Assumptions and hypotheses to be specified.• Sampling process including functional form.• Obtaining data for the analysis.• Estimation rule with good statistical properties.• Fit and test model using software package.• Analyze and evaluate implications of the results.• Problems suggest approaches for further research.

The Practice of Econometrics 1.15

Note: the textbook uses the following symbol to mark sections with advanced material:

“Skippy”

Some Basic Probability Concepts

Chapter 2

random variable: A variable whose value is unknown until it is observed.The value of a random variable results from an experiment.

The term random variable implies the existence of someknown or unknown probability distribution defined overthe set of all possible values of that variable.

In contrast, an arbitrary variable does not have aprobability distribution associated with its values.

Random Variable2.2

Controlled experiment values of explanatory variables are chosen with great care in accordance withan appropriate experimental design.

Uncontrolled experiment valuesof explanatory variables consist of nonexperimental observations overwhich the analyst has no control.

discrete random variable:A discrete random variable can take only a finitenumber of values, that can be counted by using the positive integers.

Example: Prize money from the followinglottery is a discrete random variable:

first prize: $1,000second prize: $50third prize: $5.75

since it has only four (a finite number) (count: 1,2,3,4) of possible outcomes:

$0.00; $5.75; $50.00; $1,000.00

Discrete Random Variable 2.4

continuous random variable:A continuous random variable can take any real value (not just whole numbers) in at least one interval on the real line.

Examples: Gross national product (GNP)money supplyinterest ratesprice of eggshousehold incomeexpenditure on clothing

Continuous Random Variable 2.5

A discrete random variable that is restrictedto two possible values (usually 0 and 1) iscalled a dummy variable (also, binary orindicator variable).

Dummy variables account for qualitative differences:gender (0=male, 1=female), race (0=white, 1=nonwhite),citizenship (0=U.S., 1=not U.S.), income class (0=poor, 1=rich).

Dummy Variable2.6

A list of all of the possible values takenby a discrete random variable along withtheir chances of occurring is called a probabilityfunction or probability density function (pdf).

die x f(x)one dot 1 1/6two dots 2 1/6three dots 3 1/6four dots 4 1/6five dots 5 1/6six dots 6 1/6

A discrete random variable X has pdf, f(x), which is the probabilitythat X takes on the value x.

f(x) = P(X=x)

0 < f(x) < 1

If X takes on the n values: x1, x2, . . . , xn, then f(x1) + f(x2)+. . .+f(xn) = 1.

Therefore,

Probability, f(x), for a discrete randomvariable, X, can be represented by height:

0 1 2 3 X

number, X, on Dean’s List of three roommates

f(x)0.2

A continuous random variable uses area under a curve rather than theheight, f(x), to represent probability:

X$34,000 $55,000. .

per capita income, X, in the United States

0.13240.8676

red area

green area

Since a continuous random variable has an uncountably infinite number of values, the probability of one occurring is zero.

P [ X = a ] = P [ a < X < a ] = 0

Probability is represented by area.

Height alone has no area.

An interval for X is needed to get an area under the curve.

P [ a < X < b ] = f(x) dxb

The area under a curve is the integral ofthe equation that generates the curve:

For continuous random variables it is the integral of f(x), and not f(x) itself, whichdefines the area and, therefore, the probability.

Rule 2: axi = a xi i = 1 i = 1

Rule 1: xi = x1 + x2 + . . . + xni = 1

Rule 3: xi +yi = xi + yii = 1 i = 1 i = 1

Note that summation is a linear operatorwhich means it operates term by term.

Rules of Summation2.13

Rule 4: axi +byi = a xi + b yii = 1 i = 1 i = 1

Rules of Summation (continued)

Rule 5: x = xi =i = 1

n1 x1 + x2 + . . . + xn

The definition of x as given in Rule 5 impliesthe following important fact:

xi x) = 0i = 1

Rule 6: f(xi) = f(x1) + f(x2) + . . . + f(xn)i = 1

Notation: f(xi) = f(xi) = f(xi)

x i i = 1

Rule 7: f(xi,yj) = [ f(xi,y1) + f(xi,y2)+. . .+ f(xi,ym)] i = 1 i = 1

The order of summation does not matter :

f(xi,yj) = f(xi,yj)i = 1

j = 1 j = 1

Rules of Summation (continued) 2.15

The mean or arithmetic average of arandom variable is its mathematicalexpectation or expected value, EX.

The Mean of a Random Variable

Expected Value

There are two entirely different, but mathematicallyequivalent, ways of determining the expected value:

1. Empirically: The expected value of a random variable, X,is the average value of the random variable in aninfinite number of repetitions of the experiment.

In other words, draw an infinite number of samples,and average the values of X that you get.

Expected Value

2. Analytically: The expected value of a discrete random variable, X, is determined by weighting all the possible values of X by the correspondingprobability density function values, f(x), and summing them up.

E[X] = x1f(x1) + x2f(x2) + . . . + xnf(xn)

In other words:

In the empirical case when the sample goes to infinity the values of X occur with a frequency equal to the corresponding f(x) in the analytical expression.

As sample size goes to infinity, the empirical and analytical methods will produce the same value.

Empirical vs. Analytical 2.19

x = xi

where n is the number of sample observations.

Empirical (sample) mean:

E[X] = xif(xi)i = 1

where n is the number of possible values of xi.

Analytical mean:

Notice how the meaning of n changes.

E X = xi f(xi) i=1

The expected value of X-squared:

E X = xi f(xi) i=1

It is important to notice that f(xi) does not change!

The expected value of X-cubed:

E X = xi f(xi) i=1

The expected value of X:2.21

EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)

EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)22222

= 0 + .3 + 1.2 + 1.8 + 1.6

EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) +4 (.1)33333

= 0 + .3 + 2.4 + 5.4 + 6.4

= 14.5

E [g(X)] = g(xi) f(xi)n

g(X) = g1(X) + g2(X)

E [g(X)] = g1(xi) + g2(xi)] f(xi)n

E [g(X)] = g1(xi) f(xi) + g2(xi) f(xi)n

E [g(X)] = E [g1(X)] + E [g2(X)]

Adding and Subtracting Random Variables

E(X-Y) = E(X) - E(Y)

E(X+Y) = E(X) + E(Y)

E(X+a) = E(X) + a

Adding a constant to a variable willadd a constant to its expected value:

Multiplying by constant will multiply its expected value by that constant:

E(bX) = b E(X)

var(X) = average squared deviations around the mean of X.

var(X) = expected value of the squared deviations around the expected value of X.

var(X) = E [(X - EX) ] 2

Variance2.26

var(X) = E [(X - EX) ]

= E [X - 2XEX + (EX) ]

2= E(X ) - 2 EX EX + E (EX)

= E(X ) - 2 (EX) + (EX) 2 2 2

= E(X ) - (EX) 2 2

var(X) = E [(X - EX) ] 2

var(X) = E(X ) - (EX) 22

variance of a discreterandom variable, X:

standard deviation is square root of variance

var ( X ) = (xi - EX )2 f(xi)i = 1

xi f(xi) (xi - EX) (xi - EX) f(xi)

2 .1 2 - 4.3 = -2.3 5.29 (.1) = .5293 .3 3 - 4.3 = -1.3 1.69 (.3) = .5074 .1 4 - 4.3 = - .3 .09 (.1) = .0095 .2 5 - 4.3 = .7 .49 (.2) = .0986 .3 6 - 4.3 = 1.7 2.89 (.3) = .867

xi f(xi) = .2 + .9 + .4 + 1.0 + 1.8 = 4.3

(xi - EX) f(xi) = .529 + .507 + .009 + .098 + .867 = 2.01

calculate the variance for a discrete random variable, X:

Z = a + cX

var(Z) = var(a + cX)

= E [(a+cX) - E(a+cX)]

= c var(X)

var(a + cX) = c var(X)2

A joint probability density function, f(x,y), provides the probabilities associated with the joint occurrence of all of the possible pairs of X and Y.

Joint pdf 2.31

college gradsin household

joint pdff(x,y)

Y = 1 Y = 2

vacationhomesowned

Survey of College City, NY

f(0,1) f(0,2)

f(1,1) f(1,2)

E[g(X,Y)] = g(xi,yj) f(xi,yj)i j

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75

E(XY) = xi yj f(xi,yj)i j

Calculating the expected value of functions of two random variables.

The marginal probability density functions,f(x) and f(y), for discrete random variables,can be obtained by summing over the f(x,y) with respect to the values of Y to obtain f(x) with respect to the values of X to obtain f(y).

f(xi) = f(xi,yj) f(yj) = f(xi,yj)ij

Marginal pdf 2.34

marginalY = 1 Y = 2

.50.50

f(X = 1)

f(X = 0)

f(Y = 1) f(Y = 2)

marginalpdf for Y:

marginalpdf for X:

The conditional probability density

functions of X given Y=y , f(x|y),

and of Y given X=x , f(y|x), are obtained by dividing f(x,y) by f(y)

to get f(x|y) and by f(x) to get f(y|x).

f(x|y) = f(y|x) =f(x,y) f(x,y)f(y) f(x)

Conditional pdf 2.36

conditonalY = 1 Y = 2

.50.50

.25.75

.875.125

.90.10 .70

f(Y=2|X= 0)=.25f(Y=1|X = 0)=.75

f(Y=2|X = 1)=.875

f(X=0|Y=2)=.30

f(X=1|Y=2)=.70

f(X=0|Y=1)=.90

f(X=1|Y=1)=.10

f(Y=1|X = 1)=.125

X and Y are independent random variables if their joint pdf, f(x,y),is the product of their respectivemarginal pdfs, f(x) and f(y) .

f(xi,yj) = f(xi) f(yj)for independence this must hold for all pairs of i and j

Independence 2.38

not independentY = 1 Y = 2

.50.50

f(X = 1)

f(X = 0)

f(Y = 1) f(Y = 2)

marginalpdf for Y:

marginalpdf for X:

.50x.60=.30 .50x.60=.30

.50x.40=.20 .50x.40=.20 The calculations in the boxes show the numbers required to have independence.

The covariance between two randomvariables, X and Y, measures thelinear association between them.

cov(X,Y) = E[(X - EX)(Y-EY)]

Note that variance is a special case of covariance.

cov(X,X) = var(X) = E[(X - EX) ]2

Covariance2.40

cov(X,Y) = E [(X - EX)(Y-EY)]

= E [XY - X EY - Y EX + EX EY]

= E(XY) - 2 EX EY + EX EY

= E(XY) - EX EY

cov(X,Y) = E [(X - EX)(Y-EY)]

cov(X,Y) = E(XY) - EX EY

= E(XY) - EX EY - EY EX + EX EY

Y = 1 Y = 2

.50.50

EX=0(.60)+1(.40)=.40

EY=1(.50)+2(.50)=1.50

E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75

EX EY = (.40)(1.50) = .60

cov(X,Y) = E(XY) - EX EY = .75 - (.40)(1.50) = .75 - .60 = .15

covariance

The correlation between two random variables X and Y is their covariance divided by the square roots of their respective variances.

Correlation is a pure number falling between -1 and 1.

cov(X,Y)(X,Y) =var(X) var(Y)

Correlation2.43

Y = 1 Y = 2

.50.50

EX=.40

EY=1.50

cov(X,Y) = .15

correlation

EX=0(.60)+1(.40)=.4022 2

var(X) = E(X ) - (EX) = .40 - (.40) = .24

EY=1(.50)+2(.50) = .50 + 2.0 = 2.50

var(Y) = E(Y ) - (EY) = 2.50 - (1.50) = .25

(X,Y) =cov(X,Y)

var(X) var(Y)

(X,Y) = .61

Independent random variables have zero covariance and, therefore, zero correlation.

The converse is not true.

Zero Covariance & Correlation2.45

The expected value of the weighted sumof random variables is the sum of the expectations of the individual terms.

Since expectation is a linear operator,it can be applied term by term.

E[c1X + c2Y] = c1EX + c2EY

E[c1X1+...+ cnXn] = c1EX1+...+ cnEXn

In general, for random variables X1, . . . , Xn :

The variance of a weighted sum of random variables is the sum of the variances, each times the square of the weight, plus twice the covariances of all the random variables times the products of their weights.

var(c1X + c2Y)=c1 var(X)+c2 var(Y) + 2c1c2cov(X,Y)2 2

var(c1X c2Y) = c1 var(X)+c2 var(Y) 2c1c2cov(X,Y)2 2

Weighted sum of random variables:

Weighted difference of random variables:

The Normal Distribution

Y ~ N(,2)

f(y) =2 2

(y - )2-

The Standardized Normal

Z ~ N(,)

f(z) =2

1 exp 2z2-

Z = (y - )/

P [ Y > a ] = P > = P Z > a - a - Y -

Y ~ N(,2)2.50

P [ a < Y < b ] = P < <

= P < Z <

a - Y -

Y ~ N(,2)

Y1 ~ N(1,12), Y2 ~ N(2,2

2), . . . , Yn ~ N(n,n2)

W = c1Y1 + c2Y2 + . . . + cnYn

Linear combinations of jointlynormally distributed random variablesare themselves normally distributed.

W ~ N[ E(W), var(W) ]

mean: E[V] = E[ (m) ] = m

If Z1, Z2, . . . , Zm denote m independentN(0,1) random variables, andV = Z1 + Z2 + . . . + Zm , then V ~ (m)

2 2 2 2

V is chi-square with m degrees of freedom.

Chi-Square

variance: var[V] = var[ (m) ] = 2m

If Z1, Z2, . . . , Zm denote m independentN(0,1) random variables, andV = Z1 + Z2 + . . . + Zm , then V ~ (m)

2 2 2 2

V is chi-square with m degrees of freedom.

mean: E[t] = E[t(m) ] = 0 symmetric about zero

variance: var[t] = var[t(m) ] = m / (m2)

If Z ~ N(0,1) and V ~ (m) and if Z and Vare independent then, ~ t(m)

t is student-t with m degrees of freedom.

Student - t 2.54

If V1 ~ (m1) and V2 ~ (m

2) and if V1 and V2

are independent, then~ F(m

F is an F statistic with m1 numeratordegrees of freedom and m2 denominatordegrees of freedom.

F Statistic 2.55

The Simple Linear Regression

Chapter 3

1. Estimate a relationship among economic variables, such as y = f(x).

2. Forecast or predict the value of one variable, y, based on the value of another variable, x.

Purpose of Regression Analysis

Weekly Food Expenditures

y = dollars spent each week on food items.

x = consumer’s weekly income.

The relationship between x and the expected value of y , given x, might be linear:

E(y|x) = 1 + 2 x

f(y|x=480)

yy|x=480

Figure 3.1a Probability Distribution f(y|x=480) of Food Expenditures if given income x=$480.

yy|x=480 y|x=800

Figure 3.1b Probability Distribution of FoodExpenditures if given income x=$480 and x=$800.

E(y|x)

AverageExpenditure

x (income)

E(y|x)=1+2x

2=E(y|x)

Figure 3.2 The Economic Model: a linear relationship between avearage expenditure on food and income.

xtx1=480 x2=800

y tf(yt)

Figure 3.3. The probability density function for yt at two levels of household income, x t

Homoskedastic Case

income

x tx1 x2

y tf(yt)

Figure 3.3+. The variance of yt increases as household income, x t , increases.

Heteroskedastic Case

income

Assumptions of the Simple Linear Regression Model - I

1. The average value of y, given x, is given bythe linear regression: E(y) = 1 + 2x

2. For each value of x, the values of y aredistributed around their mean with variance: var(y) = 2

3. The values of y are uncorrelated, having zerocovariance and thus no linear relationship: cov(yi ,yj) = 0

4. The variable x must take at least two differentvalues, so that x ° c, where c is a constant.

5. (optional) The values of y are normally distributed about their mean for each value of x:

y ~ N [(1+2x),2 ]

One more assumption that is often used inpractice but is not required for least squares:

The Error Term

y is a random variable composed of two parts:

I. Systematic component: E(y) = 1 + 2x This is the mean of y.

II. Random component: e = y - E(y) = y - 1 - 2x This is called the random error.

Together E(y) and e form the model: y = 1 + 2x + e

Figure 3.5 The relationship among y, e and the true regression line.

x1 x2 x3 x4

e4 E(y) = 1 + 2x

y 3.12

x1 x2 x3 x4

Figure 3.7a The relationship among y, e and the fitted regression line.

y = b1 + b2x^

x1 x2 x3 x4 x

Figure 3.7b The sum of squared residuals from any other line will be larger.

y = b1 + b2x^

y4^ y = b1 + b2x

^* * **

Figure 3.4 Probability density function for e and y

0 1+2x

1. The value of y, for each value of x, is y = 1 + 2x + e

2. The average value of the random error e is: E(e) = 0

3. The variance of the random error e is: var(e) = 2 = var(y)

4. The covariance between any pair of e’s is: cov(ei ,ej) = cov(yi ,yj) = 0

5. x must take at least two different values so that x ° c, where c is a constant.

6. e is normally distributed with mean 0, var(e)=2 (optional) e ~ N(0,2)

of the Error Term

1. Unspecified factors / explanatory variables, not in the model, may be in the error term.

2. Approximation error is in the error term if relationship between y and x is not exactly a perfectly linear relationship.

3. Strictly unpredictable random behavior that may be unique to that observation is in error.

Population regression values: y t = 1 + 2x t + e t

Population regression line: E(y t|x t) = 1 + 2x t

Sample regression values: y t = b1 + b2x t + e t

Sample regression line: y t = b1 + b2x t

y t = 1 + 2x t + e t

Minimize error sum of squared deviations:

S(1,2) = (y t - 1 - 2x t )2 (3.3.4)t=1

e t = y t - 1 - 2x t

S(1,2) = (y t - 1 - 2x t )2 (3.3.4)t =1

= - 2 (y t - 1 - 2x t )

= - 2 x t (y t - 1 - 2x t )

Set each of these two derivatives equal to zero and

solve these two equations for the two unknowns: 1 2

S(.)S(.)

Minimize w. r. t. 1 and 2:

S() = (y t - 1 - 2x t )2t =1

< 0 S(.)

> 0 S(.)i

derivatives equal to zero to get:

= - 2 (y t - b1 - b2x t ) = 0

= - 2 x t (y t - b1 - b2x t ) = 0

When these two terms are set to zero,

1 and 2 become b1 and b2 because they no longer

represent just any value of 1 and 2 but the special

values that correspond to the minimum of S() .

- 2 (y t - b1 - b2x t ) = 0

- 2 x t (y t - b1 - b2x t ) = 0

y t - Tb1 - b2 x t = 0

x t y t - b1 x t - b2 xt = 0 2

Tb1 + b2 x t = y t

b1 x t + b2 xt = x t y t 2

Solve for b1 and b2 using definitions of x and y

Tb1 + b2 x t = y t

b1 x t + b2 xt = x t y t 2

T x t yt - x t y t

T x t - ( x t)2 2

b1 = y - b2 x

elasticities

percentage change in ypercentage change in x

= = x/xy/y

= y xx y

Using calculus, we can get the elasticity at a point:

= lim = y xx y

y xx yx 0

E(y) = 1 + 2 x

E(y)x = 2

applying elasticities

estimating elasticities yx = b2

yt = b1 + b2 x t = 4 + 1.5 x t^

x = 8 = average number of years of experiencey = $10 = average wage rate

= 1.5 = 1.2810= b2

Prediction

yt = 4 + 1.5 x t^

Estimated regression equation:

x t = years of experience

yt = predicted wage rate^

If x t = 2 years, then yt = $7.00 per hour.^

If x t = 3 years, then yt = $8.50 per hour.^

log-log models

ln(y) = 1 + 2 ln(x)

ln(y)x

ln(x)x= 2

yx = 2

elasticity of y with respect to x:

Properties of Least Squares

Estimators

Chapter 4

yt = household weekly food expenditures

Simple Linear Regression Model

yt = 1 + 2 x t + t

x t = household weekly income

For a given level of x t, the expected

level of food expenditures will be:

E(yt|x t) = 1 + 2 x t

1. yt = 1 + 2x t + t

2. E( t) = 0 <=> E(yt) = 1 + 2x t

3. var( t) = 2 = var(yt)

4. cov( i, j) = cov(yi,yj) = 0

5. x t ° c for every observation

6. t~N(0,2) <=> yt~N(1+ 2x t,

Assumptions of the SimpleLinear Regression Model

are unknown population constants.

The formulas that produce thesample estimates b1 and b2 are

called the estimators of 1 and 2.

When b0 and b1 are used to represent

the formulas rather than specific values,

they are called estimators of 1 and 2which are random variables becausethey are different from sample to sample.

• If the least squares estimators b0 and b1

are random variables, then what are theirtheir means, variances, covariances andprobability distributions?

• Compare the properties of alternative estimators to the properties of the least squares estimators.

Estimators are Random Variables ( estimates are not )

The Expected Values of b1 and b2

The least squares formulas (estimators)in the simple regression case:

b2 =Txtyt - xt yt

Txt -(xt)2 2

b1 = y - b2x

where y = yt / T and x = x t / T

(3.3.8a)

(3.3.8b)

to get:

b2 = 2 +Txtt - xt t

Txt -(xt)2 2

The mean of b2 is:

Eb2 = 2 +TxtEt - xt Et

Txt -(xt)2 2

Since Et = 0, then Eb2 = 2 .

The result Eb2 = 2 means that

the distribution of b2 is centered at 2.

Since the distribution of b2

is centered at 2 ,we say that

b2 is an unbiased estimator of 2.

An Unbiased Estimator 4.8

The unbiasedness result on the previous slide assumes that weare using the correct model.

If the model is of the wrong formor is missing important variables,

then Et ° 0, then Eb2 ° 2 .

Wrong Model Specification 4.9

Unbiased Estimator of the Intercept

In a similar manner, the estimator b1

of the intercept or constant term can beshown to be an unbiased estimator of 1

when the model is correctly specified.

Eb1 = 1

b2 =Txtyt xt yt

Txt (xt)2 2 (3.3.8a)

(4.2.6)

Equivalent expressions for b2:

Expand and multiply top and bottom by T:

b2 =(xt x )yt y )xt x )

Variance of b2

Given that both yt and t have variance 2,

the variance of the estimator b2 is:

b2 is a function of the yt values butvar(b2) does not involve yt directly.

2var(b2) =

Variance of b1

x t x2var(b1) = 2

the variance of the estimator b1 is:

b1 = y b2xGiven

Covariance of b1 and b2

x t x2cov(b1,b2) = 2

If x = 0, slope can change without affectingthe variance.

1. 2: uncertainty about yt values uncertainty about b1, b2 and their relationship.

2. The more spread out the xt values are then the more confidence we have in b1, b2, etc.

3. The larger the sample size, T, the smaller the variances and covariances.

4. The variance b1 is large when the (squared) xt values are far from zero (in either direction).

5. Changing the slope, b2, has no effect on the intercept, b1, when the sample mean is zero. But if sample mean is positive, the covariance between b1 and b2 will be negative, and vice versa.

Gauss-Markov Theorm

Under the first five assumptions of the simple, linear regression model, the ordinary least squares estimators b1 and b2 have the smallest variance of all linear and unbiased estimators of1 and 2. This means that b1and b2 are the Best Linear Unbiased Estimators (BLUE) of 1 and 2.

implications of Gauss-Markov

1. b1 and b2 are “best” within the class of linear and unbiased estimators.

2. “Best” means smallest variance within the class of linear/unbiased.

3. All of the first five assumptions must hold to satisfy Gauss-Markov.

4. Gauss-Markov does not require assumption six: normality.

5. G-Markov is not based on the least squares principle but on b1 and b2.

G-Markov implications (continued)

6. If we are not satisfied with restricting our estimation to the class of linear and unbiased estimators, we should ignore the Gauss-Markov Theorem and use some nonlinear and/or biased estimator instead. (Note: a biased or nonlinear estimator could have smaller variance than those satisfying Gauss-Markov.)

7. Gauss-Markov applies to the b1 and b2

estimators and not to particular sample values (estimates) of b1 and b2.

of Least Squares Estimators

b2 ~ N 2 , x t x

b1 ~ N 1 ,x t x2

distributed The least squares estimator of 2 can beexpressed as a linear combination of yt’s:

b2 = wt yt

b1 = y b2xx t x

2where wt =x t x

This means that b1and b2 are normal sincelinear combinations of normals are normal.

normally distributed under The Central Limit Theorem

If the first five Gauss-Markov assumptionshold, and sample size, T, is sufficiently large,then the least squares estimators, b1 and b2,have a distribution that approximates thenormal distribution with greater accuracythe larger the value of sample size, T.

Consistency

We would like our estimators, b1 and b2, to collapse onto the true population values, 1 and 2, as sample size, T, goes to infinity.

One way to achieve this consistency property is for the variances of b1 and b2 to go to zero as T goes to infinity.

Since the formulas for the variances of the least squares estimators b1 and b2 show that their variances do, in fact, go to zero, then b1 and b2, are consistent estimators of 1 and 2.

of the error term, 2

et = yt b1 b2 x t^

is an unbiased estimator of

The Least Squares Predictor, yo ^

Given a value of the explanatory variable, Xo, we would like to predicta value of the dependent variable, yo.

The least squares predictor is:

yo = b1 + b2 x o (4.7.2)

Inference in the Simple

Regression Model

Chapter 5

1. yt = 1 + 2x t + t

2. E( t) = 0 <=> E(yt) = 1 + 2x t

3. var( t) = 2 = var(yt)

4. cov( i, j) = cov(yi,yj) = 0

5. x t ° c for every observation

6. t~N(0,2) <=> yt~N(1+ 2x t,

Assumptions of the Simple Linear Regression Model

Probability Distribution of Least Squares

Estimators

b1 ~ N 1 ,x t x2

2 x t2

b2 ~ N 2 , x t x2

Unbiased estimator of the error variance:

Transform to a chi-square distribution:

Error Variance Estimation 5.4

• The null hypothesis is false and we decide to reject it.

• The null hypothesis is true and we decide not to reject it.

Our decision is incorrect if:

• The null hypothesis is true and we decide to reject it. This is a type I error.

• The null hypothesis is false and we decide not to reject it. This is a type II error.

b2 ~ N 2 , x t x2

Create a standardized normal random variable, Z, by subtracting the mean of b2 and dividing by its standard deviation:

var(b2)

Simple Linear Regression

yt = 1 + 2x t + twhere E t = 0

yt ~ N(1+ 2x t , 2)

since Eyt = 1 + 2x t

t = yt 1 2x t

Therefore, t ~ N(0,2) .

Create a Chi-Square

t ~ N(0,2) but want N(0,) .

t /~ N(0,) Standard Normal .

t /~ Chi-Square .

Sum of Chi-Squares

t =1 t /=

1 / 2 / T

+. . .+ =

Therefore, t =1 t /

Since the errors t = yt 1 2x t

are not observable, we estimate them with

the sample residuals e t = yt b1 b2x t.

Unlike the errors, the sample residuals arenot independent since they use up two degreesof freedom by using b1 and b2 to estimate 1 and 2.

We get only T2 degrees of freedom instead of T.

Chi-Square degrees of freedom5.10

Student-t Distribution

t = ~ t(m)

where Z ~ N(0,1)

and V ~ (m)

t = ~ t(m)

V / ( T2)

where Z = (b2 2)

var(b2)

and var(b2) =

( xi x )2

V / (T-2)

(b2 2)

var(b2)t =

(T2) 2

^( T2)

V = (T2)

var(b2) =

( xi x )2

(b2 2)

( xi x )2

(T2) 2

^( T2)

(b2 2)

( xi x )2

notice thecancellations

(b2 2)

( xi x )2

^t = =

(b2 2)

var(b2)^

t = (b2 2)

se(b2)

Student’s t - statistic

t = ~ t (T2)

(b2 2)

se(b2)

t has a Student-t Distribution with T2 degrees of freedom.

Figure 5.1 Student-t Distribution

-tc tc

red area = rejection region for 2-sided test

probability statements

P(-tc ٹ t ٹ tc) = 1

P( t < -tc ) = P( t > tc ) =

P(-tc ٹ tc) = 1 ٹ(b2 2)

se(b2)

Confidence Intervals

Two-sided (1)x100% C.I. for 1:

b1 t/2[se(b1)], b1 + t/2[se(b1)]

b2 t/2[se(b2)], b2 + t/2[se(b2)]

Two-sided (1)x100% C.I. for 2:

Student-t vs. Normal Distribution

1. Both are symmetric bell-shaped distributions.

2. Student-t distribution has fatter tails than the normal.

3. Student-t converges to the normal for infinite sample.

4. Student-t conditional on degrees of freedom (df).

5. Normal is a good approximation of Student-t for the first few decimal places when df > 30 or so.

Hypothesis Tests1. A null hypothesis, H0.

2. An alternative hypothesis, H1.

3. A test statistic.

4. A rejection region.

Rejection Rules1. Two-Sided Test:

If the value of the test statistic falls in the critical region in either tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.

2. Left-Tail Test:If the value of the test statistic falls in the critical region which lies in the left tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.

2. Right-Tail Test:If the value of the test statistic falls in the critical region which lies in the right tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.

Format for Hypothesis Testing

1. Determine null and alternative hypotheses.

2. Specify the test statistic and its distribution as if the null hypothesis were true.

3. Select and determine the rejection region.

4. Calculate the sample value of test statistic.

5. State your conclusion.

significance in economics

Practically but not statistically significant:

When sample size is very small, a large average gap between the salaries of men and women might not be statistically significant.

Statistically but not practically significant:

When sample size is very large, a small correlation (say, = 0.00000001) between the winning numbers in the PowerBall Lottery and the Dow-Jones Stock Market Index might be statistically significant.

Type I and Type II errors

Type I error:

We make the mistake of rejecting the null hypothesis when it is true.

= P(rejecting H0 when it is true).

Type II error:

We make the mistake of failing to reject the null hypothesis when it is false. = P(failing to reject H0 when it is false).

Prediction Intervals

A (1)x100% prediction interval for yo is:

yo ± tc se( f )^

se( f ) = var( f )^f = yo yo^

x t x2var( f ) =

2 1 + +^ 1

x o x2^

The Simple Linear Regression Model

Chapter 6

Explaining Variation in yt

Predicting yt without any explanatory variables:

yt = 1 + et

et = (yt 1)2

t = 1 t = 1

= (yt b1) = 0et

2t = 1

(yt b1) = 0t = 1

yt Tb1 = 0t = 1

b1 = y

Why not y?

yt = b1 + b2xt + et^

Unexplained variation:

yt = b1 + b2xt^Explained variation:

et = yt yt = yt b1 b2xt^^

yt = yt + et^

Why not y?

yt y = yt y + et^ ^

using y as baseline

SST = SSR + SSE(yty)2 = (yty)2 +et

T ^ ^T T

t = 1 t = 1

2cross

producttermdropsout

Total Variation in yt

SST = total sum of squares

SST measures variation of yt around y

(yt y)2t = 1

Explained Variation in yt

SSR = regression sum of squares

yt = b1 + b2xt^Fitted yt values:^

SSR measures variation of yt around y^

(yt y)2t = 1

SSR = ^

Unexplained Variation in yt

SSE = error sum of squares

SSE measures variation of yt around yt^

et = ytyt = yt b1 b2xt^^

(yt yt)2 = et

2 t = 1

SSE = ^t = 1

Analysis of Variance Table

Table 6.1 Analysis of Variance Table Source of Sum of MeanVariation DF Squares Square Explained 1 SSR SSR/1Unexplained T-2 SSE SSE/(T-2) [= 2]Total T-1 SST

Coefficient of Determination

1ٹ R2 ٹ 0

What proportion of the variation in yt is explained?

SSRSSTR2 =

SST = SSR + SSE

SST SSR SSESST SST SST

SSR SSESST SST

Dividingby SST

SSRSSTR2 = = 1 SSE

R2 is only a descriptive measure.

R2 does not measure the qualityof the regression model.

Focusing solely on maximizing R2 is not a good idea.

Coefficient of Determination 6.11

cov(X,Y) =var(X) var(Y)

Correlation Analysis

cov(X,Y)r =var(X) var(Y)

Population:

Sample:

var(X) =^(xt x)2/(T1)t = 1

var(Y) =^(yt y)2/(T1)t = 1

cov(X,Y) =^(xt x)(yt y)/(T1)t = 1

(xt x)2 (yt y)2t = 1

T(xt x)(yt y)t = 1

r =t = 1

Sample Correlation Coefficient

Correlation Analysis and R2

For simple linear regression analysis:

2 = R2

R2 is also the correlation

between yt and yt

measuring “goodness of fit”.

Regression Computer Output

Table 6.2 Computer Generated Least Squares Results (1) (2) (3) (4) (5) Parameter Standard T for H0:Variable Estimate Error Parameter=0 Prob>|T|INTERCEPT 40.7676 22.1387 1.841 0.0734X 0.1283 0.0305 4.201 0.0002

Typical computer output of regression estimates:

se(b1) = var(b1) = 490.12 = 22.1287^

se(b2) = var(b2) = 0.0009326 = 0.0305^

b1 = 40.7676 b2 = 0.1283

se(b1)t = = = 1.84

b1 40.7676

22.1287

se(b2)b2t = = = 4.20

0.12830.0305

Table 6.3 Analysis of Variance Table Sum of MeanSource DF Squares Square Explained 1 25221.2229 25221.2229Unexplained 38 54311.3314 1429.2455Total 39 79532.5544 R-square: 0.3171

Sources of variation in the dependent variable:

SSRSST

R2 = = 1 = 0.317SSESST

SSE /(T-2) = 2 = 1429.2455 ^SSE = et

2 = 54311^

SST = (yty)2 = 79532

SSR = (yty)2 = 25221^

yt = 40.7676 + 0.1283xt

(s.e.) (22.1387) (0.0305)

yt = 40.7676 + 0.1283xt

(t) (1.84) (4.20)

Reporting Regression Results 6.20

R2 = 0.317

Reporting Regression Results

This R2 value may seem low but it istypical in studies involving cross-sectionaldata analyzed at the individual or micro level.

A considerably higher R2 value would beexpected in studies involving time-series dataanalyzed at an aggregate or macro level.

Changing the scale of x

yt = 1 + (c2)(xt/c) + et

yt = 1 + 2xt + et

yt = 1 + 2xt + et* *

2 = c2* xt = xt/c

The estimatedcoefficient andstandard errorchange but theother statisticsare unchanged.

Changing the scale of y

yt/c = (1/c) + (2/c)xt + et/c

yt = 1 + 2xt + et

1 = 1/c * and

All statisticsare changedexcept forthe t-statisticsand R2 value.

yt = 1 + 2xt + et* ***

2 = 2/c *

*et = et/cyt = yt/cwhere *

Changing the scale of x and y

yt/c = (1/c) + (c2/c)xt/c + et/c

yt = 1 + 2xt + et

1 = 1/c * and

No change inthe R2 or thet-statistics orin regression

results for 2

but all otherstats change.

yt = 1 + 2xt + et* ***

xt = xt/c *

*et = et/cyt = yt/cwhere *

The term linear in a simple regression model does not mean a linear relationship between variables, but a model in which the parameters enter the model in a linear way.

yt = 1 + 2xt + et

Linear Statistical Models:

Nonlinear Statistical Models:

ln(yt) = 1 + 2xt + et

yt = 1 + 2 ln(xt) + et

yt = 1 + 2xt + et2

yt = 1 + 2xt + et

yt = 1 + 2xt + exp(3xt) + et

yt = 1 + 2xt + et

Linear vs. Nonlinear 6.27

nonlinear relationship

between food expenditure and

income

Linear vs. Nonlinear

foodexpenditure

income0

Useful Functional Forms

1. Linear2. Reciprocal3. Log-Log4. Log-Linear5. Linear-Log6. Log-Inverse

Look at each form and its slope and elasticity

Linear

yt = 1 + 2xt + et

slope: 2 elasticity: 2 yt

Reciprocal

yt = 1 + 2 + et

xtslope: elasticity:

xt yt2

Log-Log

ln(yt)= 1 + 2ln(xt) + et

slope: 2 elasticity: 2

Useful Functional Forms 6.31

Log-Linear

ln(yt)= 1 + 2xt + et

slope: 2 yt elasticity: 2xt

Linear-Log

yt= 1 + 2ln(xt) + et

_slope: 2 elasticity: 2

ln(yt) = 1 - 2 + et 1xt

Log-Inverse

slope: 2 elasticity: 2x2t

yt 1xt

1. E (et) = 0

2. var (et) = 2

3. cov(ei, ej) = 0

4. et ~ N(0, 2)

Error Term Properties 6.35

Economic Models

1. Demand Models2. Supply Models3. Production Functions4. Cost Functions5. Phillips Curve

1. Demand Models

* quality demanded (yd) and price (x)* constant elasticity

Economic Models

ln(yt )= 1 + 2ln(x)t + et d

2. Supply Models

* quality supplied (ys) and price (x)* constant elasticity

Economic Models

ln(yt )= 1 + 2ln(xt) + et s

3. Production Functions* output (y) and input (x)

* constant elasticity

Economic Models

ln(yt)= 1 + 2ln(xt) + et

Cobb-Douglas Production Function:

4a. Cost Functions

* total cost (y) and output (x)

Economic Models

yt = 1 + 2x2t + et

4b. Cost Functions

* average cost (x/y) and output (x)

Economic Models

(yt/xt) = 1/xt + 2xt + et/xt

5. Phillips Curve

* wage rate (wt) and time (t)

Economic Models

unemployment rate, ut

wt-1% wt =

wt wt-1= ut

nonlinear in both variables and parameters

The Multiple Regression Model

Chapter 7

yt = 1 + 2xt2 + 3xt3 + et

= 2 xt3

yt = 3xt‘s affect yt

separately

But least squares estimation of 2

now depends upon both xt2 and xt3 .

yt = output xt2 = capital xt3 = labor

Always 5 workers per machine.

If number of workers per machine is never varied, it becomes impossible to tell if the machines or the workers are responsible for changes in output.

yt = 1 + 2xt2 + 3xt3 + et

yt = 1 + 2xt2 + 3xt3 +. . .+ KxtK + et

The parameter 1 is the intercept (constant) term.

The “variable” attached to 1 is xt1= 1.

Usually, the number of explanatory variables is said to be K1 (ignoring xt1= 1), while the

number of parameters is K. (Namely: 1 . . . K).

1. E(et) = 0

2. var(et) = 2

covet , es= for t ° s

4. et ~ N(0, 2)

Statistical Properties of et 7.5

1. E (yt) = 1 + 2xt2 +. . .+ KxtK

2. var(yt) = var(et) = 2

cov(yt ,ys) = cov(et , es) = 0 t°s

4. yt ~ N(1+2xt2 +. . .+KxtK, 2)

Statistical Properties of yt 7.6

1. yt = 1 + 2xt2 +. . .+ KxtK + et

2. E (yt) = 1 + 2xt2 +. . .+ KxtK

3. var(yt) = var(et) = 2

cov(yt ,ys) = cov(et ,es) = 0 t ° s

5. The values of xtk are not random

6. yt ~ N(1+2xt2 +. . .+KxtK, 2)

Least Squares Estimation

yt = 1 + 2xt2 + 3xt3 + et

S ؛ S(1, 2, 3) = yt12xt23xt3

Define: yt = yt y*

xt2 = xt2 x2*

xt3 = xt3 x3*

b1 = yb1b2x2 b3x3

b3 = yt xt3xt2 yt xt2xt3xt2* * * * * * *2

xt2 xt3 xt2xt3* * * *2 2 2

b2 = yt xt2xt3 yt xt3xt2xt3* * * * * * *2

xt2 xt3 xt2xt3* * * *2 2 2

Least Squares Estimators 7.9

Dangers of Extrapolation

Statistical models generally are good only“within the relevant range”. This meansthat extending them to extreme data valuesoutside the range of the original data oftenleads to poor and sometimes ridiculous results.

If height is normally distributed and the normal ranges from minus infinity to plus infinity, pity the man minus three feet tall.

Unbiased estimator of the error variance:

Transform to a chi-square distribution:

Under the assumptions of the multiple regression model, the ordinary least squares estimators have the smallest variance of all linear and unbiased estimators. This means that the least squares estimators are the Best Linear U nbiased Estimators (BLUE).

Variances

yt = 1 + 2xt2 + 3xt3 + et

var(b3) =

(1 r23)(xt3 x3)2

var(b2) =(1 r23)(xt2 x2)

(xt2 x2)2 (xt3 x3)

where r23 = (xt2 x2)(xt3 x3)

When r23 = 0these reduceto the simpleregressionformulas.

Variance Decomposition

The variance of an estimator is smaller when:

1. The error variance, 2, is smaller:

2. The sample size, T, is larger:

(xt2 x2)2 .

3. The variable’s values are more spread out: (xt2 x2)

4. The correlation is close to zero: r23 0 .

Covariances

yt = 1 + 2xt2 + 3xt3 + et

where r23 =

(xt2 x2)2 (xt3 x3)

(xt2 x2)(xt3 x3)

(1 r23) (xt2 x2)2 (xt3 x3)

cov(b2,b3) = 2

Covariance Decomposition

1. The error variance, 2, is larger.

2. The sample size, T, is smaller.

3. The values of the variables are less spread out.

4. The correlation, r23, is high.

The covariance between any two estimatorsis larger in absolute value when:

yt = 1 + 2xt2 + 3xt3 + et

var(b1) cov(b1,b2) cov(b1,b3)cov(b1,b2,b3) = cov(b1,b2) var(b2) cov(b2,b3) cov(b1,b3) cov(b2,b3) var(b3)

The least squares estimators b1, b2, and b3

have covariance matrix:

yt = 1 + 2x2t + 3x3t +. . .+ KxKt + et

yt ~N (1 + 2x2t + 3x3t +. . .+ KxKt), 2

et ~ N(0, 2)This implies and is implied by:

bk ~ N k, var(bk)

z = ~ N(0,1) for k = 1,2,...,Kbk k

var(bk)

Since bk is a linear

function of the yt’s:

var(bk)^t = =

se(bk)

Since generally the population varianceof bk , var(bk) , is unknown, we estimate it with which uses

2 instead of 2.var(bk)^ ^

t has a Student-t distribution with df=(TK).

Interval Estimation

se(bk)P tc ٹ tc = 1 ٹ

tc is critical value for (T-K) degrees of freedom

such that P(t چ tc) = /2.

P bk tc se(bk) ٹ k ٹ bk + tc se(bk) = 1

Interval endpoints: bk tc se(bk) , bk + tc se(bk)

Hypothesis Testing and

Nonsample Information

Chapter 8

1. Student-t Tests

2. Goodness-of-Fit

3. F-Tests

4. ANOVA Table

5. Nonsample Information

6. Collinearity

7. Prediction

Chapter 8: Overview 8.2

yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et

Student-t tests can be used to test any linearcombination of the regression coefficients:

H0: 2 + 3 + 4 = 1H0: 1 = 0

H0: 32 73 = 21 H0: 2 3 5ٹ

Every such t-test has exactly TK degrees of freedomwhere K=#coefficients estimated(including the intercept).

yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et

H0: 3 0ٹ

H1: 3 > 0b3

se(b3)t = ~ t (TK)

df = TK = T4

yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et

H0: 2 = 0

H1: 2 ° 0b2

se(b2)t = ~ t (TK)

df = TK = T4

1ٹ R2 ٹ 0

SSTR2 = = (yt y)2t = 1

(yt y)2t = 1

Adjusted Coefficient of Determination

Original:

Adjusted:

SST/(T1)R2 = 1 SSE/(TK)

SST= 1 SSER2 = SSTSSR

Table 8.2 Summary of Least Squares Results Variable Coefficient Std Error t-value p-value constant 104.79 6.48 16.17 0.000price 6.642 3.191 2.081 0.042advertising 2.984 0.167 17.868 0.000

se(b2)t = =

3.1912.081=

yt = Xt2 + Xt3^

(6.48) (3.191) (0.167) (s.e.)

yt = Xt2 + Xt3^

(16.17) (-2.081) (17.868) (t)

Reporting t-statistics:

Reporting standard errors:

yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et

H0: 2 = 0

H1: 2 ° 0

dfd = TK = 49dfn = J = 1

(SSER SSEU)/J

SSEU/(TK)F =

(1964.758 1805.168)/1

1805.168/(52 3)=

= 4.33

By definition this is the t-statistic squared:t = 2.081 F = t2 =

yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et

H0: 2 = 0, 4 = 0

H1: H0 not true

dfd = TK = 49dfn = J = 2

(SSER SSEU)/J

SSEU/(TK)F =

First run the restrictedregression by dropping Xt2 and Xt4 to get SSER.Next run unrestricted regression to get SSEU .

F-Tests

(SSER SSEU)/J

SSEU/(TK)F =

F-Tests of this type are always right-tailed, even for left-sided or two-sided hypotheses, because any deviation from the null will make the F value bigger (move rightward).

yt = 1 + 2Xt2 + 3Xt3 + et

H0: 2 = 3 = 0

H1: H0 not true

dfd = TK = 49dfn = J = 2

(SSER SSEU)/J

SSEU/(TK)F =

(13581.35 1805.168)/2

1805.168/(52 3)=

= 159.828

We ignore 1. Why?

Fc = 3.187

= 0.05

Reject H0!

Table 8.3 Analysis of Variance Table Sum of MeanSource DF Squares Square F-Value Explained 2 11776.18 5888.09 158.828Unexplained 49 1805.168 36.84Total 51 13581.35 p-value: 0.0001

SSTR2 = =SSR = 0.86711776.18

13581.35

ln(yt) = 1 + 2 ln(Xt2) + 3 ln(Xt3) + 4 ln(Xt4) + et

A certain production process is known to beCobb-Douglas with constant returns to scale.

2 + 3 + 4 = 1where 4 = (1 2 3)

ln(yt /Xt4) = 1 + 2 ln(Xt2/Xt4) + 3 ln(Xt3 /Xt4) + et

yt = 1 + 2 Xt2 + 3 Xt3 + 4 Xt4 + et* * * *

Run least squares on the transformed model.Interpret coefficients same as in original model.

The term “independent variable” means an explanatory variable is independent of of the error term, but not necessarily independent of other explanatory variables.

Since economists typically have no controlover the implicit “experimental design”,explanatory variables tend to movetogether which often makes sorting outtheir separate influences rather problematic.

1. no least squares output when collinearity is exact.

2. large standard errors and wide confidence intervals.

3. insignificant t-values even with high R2 and a significant F-value.

4. estimates sensitive to deletion or addition of a few observations or “insignificant” variables.

5. good “within-sample”(same proportions) but poor “out-of-sample”(different proportions) prediction.

A high degree of collinearity will produce:

Evidence of high collinearity include:

1. a high pairwise correlation between two explanatory variables.

2. a high R-squared when regressing one explanatory variable at a time on each of the remaining explanatory variables.

3. a statistically significant F-value when the t-values are statistically insignificant.

4. an R-squared that doesn’t fall by much when dropping any of the explanatory variables.

Since high collinearity is not a violation ofany least squares assumption, but rather a lack of adequate information in the sample:

1. collect more data with better information.

2. impose economic restrictions as appropriate.

3. impose statistical restrictions when justified.

4. if all else fails at least point out that the poor model performance might be due to the collinearity problem (or it might not).

Given a set of values for the explanatoryvariables, (1 X02 X03), the best linearunbiased predictor of y is given by:

yt = 1 + 2Xt2 + 3Xt3 + et

This predictor is unbiased in the sensethat the average value of the forecasterror is zero.

y0 = b1 + b2X02 + b3X03^

Extensions of the Multiple

Regression Model

Chapter 9

Topics for This Chapter

1. Intercept Dummy Variables

2. Slope Dummy Variables

3. Different Intercepts & Slopes

4. Testing Qualitative Effects

5. Are Two Regressions Equal?

6. Interaction Effects

7. Dummy Dependent Variables

Intercept Dummy Variables

Dummy variables are binary (0,1)

Dt = 1 if red car, Dt = 0 otherwise.

yt = 1 + 2Xt + 3Dt + et

yt = speed of car in miles per hour

Xt = age of car in years

Police: red cars travel faster.H0: 3 = 0H1: 3 > 0

yt = 1 + 2Xt + 3Dt + et

red cars: yt = (1 + 3) + 2xt + et other cars: yt = 1 + 2Xt + et

milesper hour

age in years0

red cars

other cars

yt = 1 + 2Xt + 3DtXt + et

yt = 1 + (2 + 3)Xt + et

yt = 1 + 2Xt + et

valueofporfolio

years0

stocks

Stock portfolio: Dt = 1 Bond portfolio: Dt = 0

1 = initial

investment

Different Intercepts & Slopes

yt = 1 + 2Xt + 3Dt + 4DtXt + et

yt = (1 + 3) + (2 + 4)Xt + et

yt = 1 + 2Xt + et

harvestweightof corn

rainfall

“miracle”

regular

“miracle” seed: Dt = 1 regular seed: Dt = 0

yt = 1 + 2 Xt + et

For men Dt = 1. For women Dt = 0.

years of experience

yt = (1+ 3) + 2 Xt + et

wagerate

H0: 3 = 0

H1: 3 > 0 .

. Testing fordiscriminationin starting wage

yt = 1 + (5 +6 )Xt + et

yt = 1 + 5 Xt + et

For men Dt = 1.

For women Dt = 0.

Men and women have the same starting wage, 1 , but their wage ratesincrease at different rates (diff.= 6 ).

6 > means that men’s wage rates areincreasing faster than women's wage rates.

years of experience

wagerate

yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et

yt = (1 + 3) + (2 + 4) Xt + et

yt = 1 + 2 Xt + et

Women are given a higher starting wage, 1 , while men get the lower starting wage, 1 + 3 ,(3 < 0 ). But, men get a faster rate of increasein their wages, 2 + 4 , which is higher than therate of increase for women, 2 , (since 4 > 0 ).

years of experience

An Ineffective Affirmative Action Plan

women are startedat a higher wage.

Note:(3 < 0 )

wagerate

Testing Qualitative Effects

1. Test for differences in intercept.

2. Test for differences in slope.

3. Test for differences in both intercept and slope.

H0: vs1:

Yt 12Xt3Dt

4Dt Xt

Est. Var b3کt n 4

Est. Var b4کt n 4

men: Dt = 1 ; women: Dt = 0

Testing fordiscrimination instarting wage.

Testing fordiscrimination inwage increases.

intercept

Testing Ho:

H1 : otherwise

SSE R yt b 1 b 2 X t 2

SSE U yt b1bXt bDt b Dt Xt2

SSER SSEU 2

SSEU T 4 F T 4

intercept and slope

Are Two Regressions Equal?

yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et

variations of “The Chow Test”

I. Assuming equal variances (pooling):

men: Dt = 1 ; women: Dt = 0

Ho: 3 = 4 = 0 vs. H1: otherwise

yt = wage rate

This model assumes equal wage rate variance.

Xt = years of experience

yt = 1 + 2 Xt + et

II. Allowing for unequal variances:

ytm = 1 + 2 Xtm + etm

ytw = 1 + 2 Xtw + etw

Everyone:

Men only:Women only:

Forcing men and women to have same 1, 2.

Allowing men and women to be different.SSEm

where SSEU = SSEm + SSEw

F =(SSER SSEU)/J

SSEU /(TK)

J = # restrictions

K=unrestricted coefs.

(running three regressions)

J = 2 K = 4

Interaction Variables

1. Interaction Dummies

2. Polynomial Terms (special case of continuous interaction)

3. Interaction Among Continuous Variables

1. Interaction Dummies

yt = 1 + 2 Xt + 3 Mt + 4 Bt + et

For men Mt = 1. For women Mt = 0. For black Bt = 1. For nonblack Bt = 0.

No Interaction: wage gap assumed the same:

yt = 1 + 2 Xt + 3 Mt + 4 Bt + 5 Mt Bt + et

Interaction: wage gap depends on race:

Wage Gap between Men and Women

yt = wage rate; Xt = experience

2. Polynomial Terms

yt = 1 + 2 X t + 3 X2

t + 4 X3

t + et

Linear in parameters but nonlinear in variables:

yt = income; Xt = agePolynomial Regression

X tPeople retire at different ages or not at all.

9020 30 40 50 60 8070

yt = 1 + 2 X t + 3 X2

t + 4 X3

t + et

yt = income; Xt = age

Polynomial Regression

Rate income is changing as we age:yt

= 2 + 2 3 X t + 3 4 X

Slope changes as X t changes.

3. Continuous Interaction

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et

Exam grade = f(sleep:Zt , study time:Bt)

Sleep and study time do not act independently.

More study time will be more effective when combined with more sleep and less effective when combined with less sleep.

Your mind sortsthings out whileyou sleep (when you have things to sort out.)

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et

= 2 + 4 Zt

Your studying is more effectivewith more sleep.

= 2 + 4 Bt

continuous interaction 9.20

yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et

If Zt + Bt = 24 hours, then Bt = (24 Zt)

yt = 1+ 2 Zt +3(24 Zt) +4 Zt (24 Zt) + et

yt = (1+24 3) + (23+24 4)Zt 4Z2

t + et

yt = 1 + 2 Zt + 3 Z2

t + et

Sleep needed to maximize your exam grade:yt

= 2 + 23 Zt = 0where 2 > 0 and 3 < 0

1. Linear Probability Model

2. Probit Model

3. Logit Model

Dummy Dependent Variables9.22

Linear Probability Model

yi = 1 + 2 Xi2 + 3 Xi3 + 4 Xi4 + ei

Xi2 = total hours of work each week

1 quits job 0 does not quit

Xi3 = weekly paycheck

Xi4 = hourly pay (Xi3 divided by Xi2)

yi = 1 + 2 Xi2 + 3 Xi3 + 4 Xi4 + ei

yt = 1

0yt =total hours of work each week

yi = b1 + b2 Xi2 + b3 Xi3 + b4 Xi4^

Read predicted values of yi off the regression line

Linear Probability Model 9.24

1. Probability estimates are sometimesless than zero or greater than one.

2. Heteroskedasticity is present in that the model generates a nonconstant error variance.

Linear Probability Model

Problems with Linear Probability Model:

Probit Model

zi = 1 + 2 Xi2 +

2 f(zi) = e0.5zi

F(zi) = P[ Z zi ] = e0.5u2

du 2 1

Normal probability density function:

Normal cumulative probability function:

latent variable, zi :

pi = P[ Z 1 + 2Xi2 ] = F(1 + 2Xi2)

Since zi = 1 + 2 Xi2 + , we cansubstitute in to get

Probit Model

Xi2total hours of work each week

yt = 1

Logit Model

1 + e (1 +

Define pi :

For 2 > 0, pi will approach 1 as Xi2 +

pi is the probability of quitting the job.

For 2 > 0, pi will approach 0 as Xi2

Logit Model

Xi2total hours of work each week

yt = 1

1 + e (1 +

pi is the probability of quitting the job.

Maximum Likelihood

Maximum likelihood estimation (MLE)is used to estimate Probit and Logit functions.

The small sample properties of MLE are not known, but in large samples MLE is normally distributed, and it is consistent and asymptotically efficient.

Heteroskedasticity

Chapter 10

The Nature of Heteroskedasticity

Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant.

Ordinary least squares assumes that all observations are equally reliable.

For efficiency (accurate estimation/prediction) reweight observations to ensure equal error variance.

yt = 1 + 2xt + et

Regression Model

E(et) = 0

var(et) = 2

zero mean:

homoskedasticity:

nonautocorrelation: cov(et, es) = t ° s

heteroskedasticity: var(et) = t2

Homoskedastic pattern of errors

. .. ... . .

. .. .. .

income

consumption

xtx1 x2

y tf(yt)

The Homoskedastic Case

x3 x4 income

Heteroskedastic pattern of errors

. .. .

..... ..... .

income

consumption

x tx1 x2

y tf(yt)

The Heteroskedastic Case

income

rich people

poor people

Properties of Least Squares

1. Least squares still linear and unbiased.

2. Least squares not efficient.

3. Usual formulas give incorrect standard errors for least squares.

4. Confidence intervals and hypothesis tests based on usual standard errors are wrong.

yt = 1 + 2xt + et

heteroskedasticity: var(et) = t2

incorrect formula for least squares variance:

var(b2) = 2

correct formula for least squares variance:

var(b2) = t2xt x

Hal White’s Standard Errors

White’s estimator of the least squares variance:

est.var(b2) = et2xt x

In large samples White’s standard error (square root of estimated variance) is acorrect / accurate / consistent measure.

Two Types of Heteroskedasticity

1. Proportional Heteroskedasticity. (continuous function(of xt, for example))

2. Partitioned Heteroskedasticity. (discrete categories/groups)

Proportional Heteroskedasticity

yt = 1 + 2xt + et

var(et) = t2E(et) = 0 cov(et, es) = 0 t ° s

t2 = 2 xt

The variance is assumed to be proportional to the value of xt

t2 = 2 xt

yt = 1 + 2xt + et

std.dev. proportional to xt

variance:

standard deviation: t = xt

yt 1 xt et= 1 + 2 + xt xt xt xt

To correct for heteroskedasticity divide the model by xt

var(et) = t2

yt = 1xt1 + 2xt2 + et* * * *

var(et ) = var( ) = var(et) = 2 xt* et

var(et ) = 2*

et is heteroskedastic, but et is homoskedastic.*

1. Decide which variable is proportional to the heteroskedasticity (xt in previous example).

2. Divide all terms in the original model by the square root of that variable (divide by xt ).

3. Run least squares on the transformed model which has new yt, xt1 and xt2 variables

but no intercept.

Generalized Least Squares

These steps describe weighted least squares:

Partitioned Heteroskedasticity

yt = 1 + 2xt + et

var(et) = 12

var(et) = 22

error variance of “field” corn:

error variance of “sweet” corn:

yt = bushels per acre of corn

xt = gallons of water per acre (rain or other)

t = 1, . . . ,100

t = 1, . . . ,80

t = 81, . . . ,100

yt = 1 + 2xt + et var(et) = 12“field” corn:

yt = 1 + 2xt + et var(et) = 22“sweet” corn:

yt 1 xt et= 1 + 2 + 1 1 1 1

yt 1 xt et= 1 + 2 + 2 2 2 2

Reweighting Each Group’s Observations

t = 1, . . . ,80

t = 81, . . . ,100

Apply Generalized Least Squares

Run least squares separately on data for each group.

12 provides estimator of 1

2 using

the 80 observations on “field” corn.

22 provides estimator of 2

2 using

the 20 observations on “sweet” corn.

1. Residual Plots provide information on the exact nature of heteroskedasticity (partitioned or proportional) to aid in correcting for it.

2. Goldfeld-Quandt Test checks for presence of heteroskedasticity.

Detecting Heteroskedasticity

Determine existence and nature of heteroskedasticity:

Residual Plots

. .. ..

. . ..

Plot residuals against one variable at a timeafter sorting the data by that variable to tryto find a heteroskedastic pattern in the data.

Goldfeld-Quandt Test

The Goldfeld-Quandt test can be used to detect heteroskedasticity in either the proportional case or for comparing two groups in the discrete case.

For proportional heteroskedasticity, it is first necessaryto determine which variable, such as xt, is proportionalto the error variance. Then sort the data from the largest to smallest values of that variable.

Ho: 12 = 2

H1: 12 > 2

GQ = ~ F[T1-K1, T2-K2]

In the proportional case, drop the middle r observations where r T/6, then runseparate least squares regressions on the firstT1 observations and the last T2 observations.

Small values of GQ support Ho while large values support H1.

Goldfeld-Quandt Test Statistic

Use F Table

t2 = 2 exp{1 zt1 + 2 zt2}

More General Model

Structure of heteroskedasticity could be more complicated:

zt1 and zt2 are any observable variables upon

which we believe the variance could depend.

Note: The function exp{.} ensures that t2 is positive.

t2 = 2 exp{1 zt1 + 2 zt2}

More General Model

lnt2 = ln

2+ 1 zt1 + 2 zt2

lnt2 = + 1 zt1 + 2 zt2

where = ln2

Ho: 1 = 0, 2

H1: 1 ° 0, 2

° 0 and/or

Least squares residuals, et ^

lnet2 =+1zt1+2zt2 + t

the usual F test

Autocorrelation

Chapter 11

The Nature of Autocorrelation

For efficiency (accurate estimation/prediction) all systematic information needs to be incor-porated into the regression model.

Autocorrelation is a systematic pattern in the errors that can be either attracting (positive) or repelling (negative) autocorrelation.

PostiveAuto.

NoAuto.

NegativeAuto.

.. . . .

. . . . .. .

. . . ... ...

.. ... .

.. .. ... .

crosses line not enough (attracting)

crosses line randomly

crosses line too much (repelling)

yt = 1 + 2xt + et

Regression Model

E(et) = 0

var(et) = 2

zero mean:

homoskedasticity:

nonautocorrelation: cov(et, es) = t ° s

autocorrelation: cov(et, es) ° t ° s

Order of Autocorrelation

yt = 1 + 2xt + et

et = et1 + t

et = 1et1 + 2et2 + t

et = 1et1 + 2et2 + 3et3 + t

1st Order:

2nd Order:

3rd Order:

We will assume First Order Autocorrelation:

et = et1 + tAR(1) :

First Order Autocorrelation

yt = 1 + 2xt + et

et = et1 + t where 1 < < 1

E(t) = 0 var(t) = 2 cov(t, s) = t ° s

These assumptions about t imply the following about et :

E(et) = 0

var(et) = e2 =

cov(et, etk) = e2 k for k > 0

corr(et, etk) = k for k > 0

Autocorrelation creates someProblems for Least Squares:

1. The least squares estimator is still linear and unbiased but it is not efficient.

2. The formulas normally used to compute the least squares standard errors are no longer correct and confidence intervals and hypothesis tests using them will be wrong.

Generalized Least Squares

yt = 1 + 2xt + et

et = et1 + t

yt = 1 + 2xt + et1 + t

substitute in for et

Now we need to get rid of et1

(continued)

AR(1) :

yt = 1 + 2xt + et

yt = 1 + 2xt + et1 + t

et = yt 12xt

et1 = yt1 12xt1

yt = 1 + 2xt + yt1 12xt1 + t

lag theerrorsonce

(continued)

yt = 1 + 2xt + yt1 12xt1 + t

yt yt1 = 1(1) + 2(xtxt1) + t

yt = 1 + 2xt2 + t* * *

yt = yt yt1 *

1 = 1(1) *

xt2 = (xtxt1) *

yt = 1 + 2xt2 + t* * *

yt = yt yt1 * 1 = 1(1) *

xt2 = xt xt1*

Problems estimating this model with least squares:

1. One observation is used up in creating the transformed (lagged) variables leaving only (T1) observations for estimating the model.

2. The value of is not known. We must find some way to estimate it.

Recovering the 1st Observation

Dropping the 1st observation and applying least squares is not the best linear unbiased estimation method.

Efficiency is lost because the varianceof the error associated with the 1st observation is not equal to that of the other errors.

This is a special case of the heteroskedasticityproblem except that here all errors are assumedto have equal variance except the 1st error.

Recovering the 1st Observation

y1 = 1 + 2x1 + e1

The 1st observation should fit the original model as:

We could include this as the 1st observation for ourestimation procedure but we must first transform it sothat it has the same error variance as the other observations.

with error variance: var(e1) = e2 =

2 /(1-2).

Note: The other observations all have error variance 2.

y1 = 1 + 2x1 + e1

with error variance: var(e1) = e2 =

2 /(1-2).

The other observations all have error variance 2.

Given any constant c : var(ce1) = c2 var(e1).

If c = 1-2 , then var( 1-2 e1) = (1-2) var(e1).

= (1-2) e2

= (1-2) 2 /(1-2)

The transformation 1 = 1-2 e1 has variance 2 .

y1 = 1 + 2x1 + e1

The transformed error 1 = 1-2 e1 has variance 2 .

Multiply through by 1-2 to get:

1-2 y1 = 1-2 1 + 1-2 2x1 + 1-2 e1

This transformed first observation may now be added to the other (T-1) observations to obtain the fully restored set of T observations.

Estimating Unknown Value

et = et1 + t

First, use least squares to estimate the model:

If we had values for the et’s, we could estimate:

yt = 1 + 2xt + et

The residuals from this estimation are:

et = yt - b1 - b2xt^

et = et1 + t^ ^ ^

Next, estimate the following by least squares:

The least squares solution is:

et et-1

Durbin-Watson Test

Ho: = 0 vs. H1: ° 0 , > 0, or < 0

et et-1

The Durbin-Watson Test statistic, d, is :

Testing for AutocorrelationThe test statistic, d, is approximately related to as:^

d 2(1)^

When = 0 , the Durbin-Watson statistic is d 2.^

When = 1 , the Durbin-Watson statistic is d 0.^

Tables for critical values for d are not always readily available so it is easier to use the p-valuethat most computer programs provide for d.

Reject Ho if p-value < , the significance level.

Prediction with AR(1) Errors

When errors are autocorrelated, the previous period’s error may help us predict next period’s error.

The best predictor, yT+1 , for next period is:

yT+1 = 1 + 2xT+1 + eT^ ^ ^ ^ ~

where 1 and 2 are generalized least squares

estimates and eT is given by:~

eT = yT 1 2xT ^ ^ ~

yT+h = 1 + 2xT+h + h eT^ ^ ^ ^ ~

For h periods ahead, the best predictor is:

Assuming | | < 1, the influence of h eT

diminishes the further we go into the future(the larger h becomes).

Pooling

Time-Series and

Cross-Sectional Data

Chapter 12

Pooling Time and Cross Sections

yit = 1it + 2itx2it + 3itx3it + eit

If left unrestricted,this model requires different equationsfor each firm in each time period.

for the ith firm in the tth time period

Seemingly Unrelated Regressions

yit = 1i + 2ix2it + 3ix3it + eit

SUR models impose the restrictions:

1it = 1i 2it = 2i 3it = 3i

Each firm gets its own coefficients: 1i , 2i and 3i

but those coefficients are constant over time.

The investment expenditures (INV) of General Electric (G) and Westinghouse(W) may be related to their stock marketvalue (V) and actual capital stock (K) as follows:

INVGt = 1G + 2GVGt + 3GKGt + eGt

INVWt = 1W + 2WVWt + 3WKWt + eWt

i = G, W t = 1, . . . , 20

Two-Equation SUR Model12.4

Estimating Separate Equations

For now make the assumption of no correlationbetween the error terms across equations:

We make the usual error term assumptions:

cov(eGt, eGs) = 0 cov(eWt, eWs) = 0

var(eGt) = G2

var(eWt) = W2

E(eGt) = 0 E(eWt) = 0

cov(eGt, eWt) = 0 cov(eGt, eWs) = 0

INVt = 1G + 1Dt + 2GVt + 2DtVt + 3GKt + 3DtKt + et

Dummy variable model assumes that :G = W

For Westinghouse observations Dt = 1; otherwise Dt = 0.

1W = 1G + 1

2W = 2G + 2

3W = 3G + 3

Problem with OLS on Each Equation

The first assumption of the Gauss-Markov Theorem concerns the model specification.

If the model is not fully and correctly specified the Gauss-Markov properties might not hold.

Any correlation of error terms across equations must be part of model specification.

Any correlation between the dependent variables of two or more equations that is not due to their explanatory variables is by default due to correlated error terms.

Correlated Error Terms12.8

1. Sales of Pepsi vs. sales of Coke.(uncontrolled factor: outdoor temperature)

2. Investments in bonds vs. investments in stocks. (uncontrolled factor: computer/appliance sales)

3. Movie admissions vs. Golf Course admissions. (uncontrolled factor: weather conditions)

4. Sales of butter vs. sales of bread. (uncontrolled factor: bagels and cream cheese)

Which of the following models wouldbe likely to produce positively correlatederrors and which would produce negatively correlations errors?

Joint Estimation of the Equations

INVGt = 1G + 2GVGt + 3GKGt + eGt

INVWt = 1W + 2WVWt + 3WKWt + eWt

cov(eGt, eWt) = GW

Seemingly Unrelated Regressions

When the error terms of two or more equationsare correlated, efficient estimation requires the useof a Seemingly Unrelated Regressions (SUR) type estimator to take the correlation into account.

Be sure to use the Seemingly Unrelated Regressions (SUR)procedure in your regression software program to estimateany equations that you believe might have correlated errors.

Separate vs. Joint Estimation

SUR will give exactly the same results as estimating each equation separately with OLS if either or both of the following two conditions are true:

1. Every equation has exactly the same set of explanatory variables with exactly the same values.

2. There is no correlation between the error terms of any of the equations.

Test for Correlation

Test the null hypothesis of zero correlation

^ ^rGW 2

2 = T rGW

from eachequation estimatedseparately.

^ ^rGW 2

= T rGW

2 2(1)

GW eGteWt1T

G eGt1T

^^ 2 2

W eWt1T

^^ 2 2

Fixed Effects Model

yit = 1it + 2itx2it + 3itx3it + eit

yit = 1i + 2x2it + 3x3it + eit

Fixed effects models impose the restrictions:

1it = 1i 2it = 2 3it = 3

For each ith cross section in the tth time period:

Each ith cross-section has its own constant 1i intercept.

The Fixed Effects Model is conveniently represented using dummy variables:

yit = 11D1i + 12D2i + 13D3i + 14D4 i+ 2x2it + 3x3it + eit

D1i=1 if NorthD1i=0 if not N

D2i=1 if EastD2i=0 if not E

D3i=1 if SouthD3i=0 if not S

D4i=1 if WestD4i=0 if not W

yit = millions of bushels of corn producedx2it = price of corn in dollars per bushelx3it = price of soybeans in dollars per bushel

Each cross-sectional unit gets its own intercept,but each cross-sectional intercept is constant over time.

Ho : 11 = 12 = 13 = 14

Test for Equality of Fixed Effects

H1 : Ho not true

The Ho joint null hypothesis may be tested with F-statistic:

(SSER SSEU) / J

SSEU / (NT K)F = ~ F

(NT K)

SSER is the restricted error sum of squares (one intercept)SSEU is the unrestricted error sum of squares (four intercepts)N is the number of cross-sectional units (N = 4)K is the number of parameters in the model (K = 6)J is the number of restrictions being tested (J = N1 = 3)T is the number of time periods

Random Effects Model

1i = 1 + i

1 is the population mean intercept.

i is an unobservable random error thataccounts for the cross-sectional differences.

1i = 1 + i

i are independent of one another and of eit

E(i) = 0 var(i) = 2

where i = 1, ... ,N

Consequently, E(1i) = 1 var(1i) = 2

Random Intercept Term 12.19

yit = (1+i) + 2x2it + 3x3it + eit

yit = 1 + 2x2it + 3x3it + (i +eit)

yit = 1 + 2x2it + 3x3it + it

Random Effects Model 12.20

it = (i +eit)

yit = 1 + 2x2it + 3x3it + it

it has zero mean: E(it) = 0

it is homoskedastic: var(it) =+ e2 2

The errors from the same firm in different time periodsare correlated:

The errors from different firms are always uncorrelated:

cov(it,is) =2

cov(it,js) =

t ° s

i ° j

Simultaneous

Equations

Models

Chapter 13

Keynesian Macro Model

Assumptions of Simple Keynesian Model

1. Consumption, c, is function of income, y.

2. Total expenditures = consumption + investment.

3. Investment assumed independent of income.

consumption is a function of income:

income is either consumed or invested:

c = 1 + 2 y

y = c + i

The Structural Equations 13.3

The Statistical Model

ct = 1 + 2 yt + et

yt = ct + it

The consumption equation:

The income identity:

ct = 1 + 2 yt + et

yt = ct + it

Since yt

contains et

they arecorrelated

The Failure of Least Squares

The least squares estimators of parameters in a structural simul-taneous equation is biased andinconsistent because of the cor-relation between the random errorand the endogenous variables onthe right-hand side of the equation.

Single Equation: Simultaneous Equations:

Single vs. Simultaneous Equations

yt i t

Deriving the Reduced Form

ct = 1 + 2 yt + et

yt = ct + it

ct = 1 + 2(ct + it) + et

(1 2)ct = 1 + 2 it + et

ct = + it + et(12) (12) (12)11

ct = 11 + 21 it + t

The Reduced Form Equation

Reduced Form Equation

ct = 11 + 21 it + t

(12)111 = (12)

(12)1t = + et and

yt = ct + itwhere ct = 11 + 21 it + t

yt = 12 + 22 it + t

It is sometimes useful to give this equationits own reduced form parameters as follows:

yt = 11 + (1+21) it + t

yt = 12 + 22 it + t

ct = 11 + 21 it + t

Since ct and yt are related through the identity:

yt = ct + it , the error term, t, of these two

equations is the same, and it is easy to show that:

(12)111 = 12 =

(12)22 = (121) = 1

IdentificationThe structural parameters are 1 and 2.

The reduced form parameters are 11 and 21.

Once the reduced form parameters are estimated,the identification problem is to determine if theorginal structural parameters can be expresseduniquely in terms of the reduced form parameters.

(121)2 =

^(121)1 =

Identification

An equation is exactly identified if its structural (behavorial) parameters can be uniquely expres-sed in terms of the reduced form parameters.

An equation is over-identified if there is morethan one solution for expressing its structural (behavorial) parameters in terms of the reducedform parameters.

An equation is under-identified if its structural (behavorial) parameters cannot be expressed in terms of the reduced form parameters.

A system of M equations containing M endogenous variables must exclude at least M1 variables from a given equation in order for the parameters of that equation to be identified and to be able tobe consistently estimated.

Two Stage Least Squares

Problem: right-hand endogenous variablesyt2 and yt1 are correlated with the error terms.

yt1 = 1 + 2 yt2 + 3 xt1 + et1

yt2 = 1 + 2 yt1 + 3 xt2 + et2

Solution: First, derive the reduced form equations.

yt1 = 1 + 2 yt2 + 3 xt1 + et1

yt2 = 1 + 2 yt1 + 3 xt2 + et2

yt1 = 11 + 21 xt1 + 31 xt2 + t1

yt2 = 12 + 22 xt1 + 32 xt2 + t2

Solve two equations for two unknowns, yt1, yt2 :

yt1 = 11 + 21 xt1 + 31 xt2 + t1

yt2 = 12 + 22 xt1 + 32 xt2 + t2

Use least squares to get fitted values:

2SLS: Stage I

yt1 = 11 + 21 xt1 + 31 xt2^ ^ ^ ^

yt2 = 12 + 22 xt1 + 32 xt2^ ^ ^ ^ yt2 = yt2 + t2

^ ^yt1 = yt1 + t1

yt2 = yt2 + t2^ ^yt1 = yt1 + t1

^ ^ and

yt1 = 1 + 2 yt2 + 3 xt1 + et1

yt2 = 1 + 2 yt1 + 3 xt2 + et2

Substitue in

for yt1 , yt2

yt1 = 1 + 2 (yt2 + t2) + 3 xt1 + et1^ ^

yt2 = 1 + 2 (yt1 + t1) + 3 xt2 + et2^^

yt1 = 1 + 2 yt2 + 3 xt1 + ut1^

yt2 = 1 + 2 yt1 + 3 xt2 + ut2^

^ut1 = 2t2 + et1 ut2 = 2t1 + et2^where and

Run least squares on each of the above equationsto get 2SLS estimates:

1 , 2 , 3 , 1 , 2 and 3

~ ~ ~ ~ ~ ~

Nonlinear

Squares

Chapter 14

(A.) “Regression” model with only an intercept term:

Review of Least Squares Principle

yt = + et

et = yt

et = (yt )2 2

SSE = (yt )2

SSE = 2 (yt ) = 0^

yt = 0^

= yt = y^ 1T

(minimize the sum of squared errors)

Yields an exact analytical solution:

Review of Least Squares(B.) Regression model without an intercept term:

yt = xt + et

et = yt xt

et = (yt xt)2 2

SSE = (yt xt)2

SSE = 2 xt(yt xt)= 0^

xtyt xt = 0^ 2

xt yt xt = 0^ 2

xt = xtyt^ 2

=^ xtyt

This yields an exact analytical solution:

yt = + xt + et SSE = (yt xt)2

SSE = 2 (yt xt) = 0^ ^

SSE = 2 xt(yt xt) = 0^ ^

y x = 0^ ^

=^ (xtx)(yty)

(xtx)2xtyt xt xt = 0^ ^ 2

This yields an exact analytical solution:

= y x^ ^

yt = xt + et

SSE = (yt xt)

PROBLEM: An exactanalytical solution tothis does not exist.

SSE = 2 xt

ln(xt)(yt xt) = 0

xtln(xt)yt] xt

ln(xt)] = 0 ^^

Must use numericalsearch algorithm totry to find value of to satisfy this.

Find Minimum of Nonlinear SSE

SSE SSE = (yt xt)

The least squares principle is still appropriate when themodel is nonlinear, but it is harder to find the solution.

Conclusion14.7

Nonlinear least squares optimization methods:

The Gauss-Newton Method

Optional Appendix 14.8

The Gauss-Newton Algorithm

1. Apply the Taylor Series Expansion to the nonlinear model around some initial b(o).

2. Run Ordinary Least Squares (OLS) on the linear part of the Taylor Series to get b(m).

3. Perform a Taylor Series around the new b(m) to get b(m+1) .

4. Relabel b(m+1) as b(m) and rerun steps 2.-4.

5. Stop when (b(m+1) b(m) ) becomes very small.

yt = f(Xt,b) + t for t = 1, . . . , n.

Do a Taylor Series Expansion around the vector b = b(o) as follows:

yt = f(Xt,b) + f’(Xt,b)(b - b) + t

where t؛ (b - b(o))

Tf’’(Xt,b)(b - b) + Rt + t

f(Xt,b) = f(Xt,b) + f’(Xt,b)(b - b)

+ (b - b)Tf’’(Xt,b)(b - b) + Rt

yt - f(Xt,b) = f’(Xt,b)b - f’(Xt,b) b + t

yt - f(Xt,b) + f’(Xt,b) b = f’(Xt,b)b + t

yt = f’(Xt,b)b + t

where yt

؛yt - f(Xt,b) + f’(Xt,b) b

This is linear in b .

Gauss-Newton just runs OLS on thistransformed truncated Taylor series.

yt = f’(Xt,b)b + t

Gauss-Newton just runs OLS on thistransformed truncated Taylor series.

or y = f’(X,b)b + خ

for t = 1, . . . , n in matrix terms

b [ f’(X,b)T f’(X,b)]-1 f’(X,b)

This is analogous to linear OLS where y = Xb + خ led to the solution: b XTX)XTy^except that X is replaced with the matrix of firstpartial derivatives: f’(Xt,b) and y is replaced by y

(i.e. “y” = y* and “X” = f’(X,b) )

Recall that: y*(o)

yf(X,b(o)) + f’(X,b) b؛

Now define: y

؛ y f(X,b(o))

Therefore: y

= y + f’(X,b) b

b [ f’(X,b)T f’(X,b)]-1 f’(X,b)

Now substitute in for yin Gauss-Newton solution:

to get:

b = b(o) + [ f’(X,b)T f’(X,b)]-1 f’(X,b)

b(1) = b+ [ f’(X,b)T f’(X,b)]-1 f’(X,b)

Now call this b value bas follows:^

More generally, in going from interation m toiteration (m+1) we obtain the general expression:

b(m+1) = b(m) + [ f’(X,b(m)T f’(X,b(m)]-1 f’(X,b(m)

T y(m)

b(m+1) = [ f’(X,b(m)T f’(X,b(m)]-1 f’(X,b(m))

T y*(m)

b(m+1) = b(m) + [ f’(X,b(m))T f’(X,b(m))]-1 f’(X,b(m))

T y(m)

Thus, the Gauss-Newton (nonlinear OLS) solutioncan be expressed in two alternative, but equivalent,forms:

1. replacement form:

2. updating form:

For example, consider Durbin’s Method of estimatingthe autocorrelation coefficient under a first-order autoregression regime:

y t = b1 + b2Xt 2 + . . . + bK Xt K + t for t = 1, . . . , n.

t = t - 1 + ut where u t satisfies the conditions

E u t = 0 , E u 2t = su2, E u t u s = 0 for s ° t.

Therefore, u t is nonautocorrelated and homoskedastic.

Durbin’s Method is to set aside a copy of the equation,lag it once, multiply by and subtract the new equation

from the original equation, then move the yt-1 term to

the right side and estimate along with the bs by OLS.

Durbin’s Method is to set aside a copy of the equation,lag it once, multiply by and subtract the new equation

from the original equation, then move the yt-1 term to

the right side and estimate along with the b’s by OLS.

y t = b1 + b2Xt 2 + b3 X t 3 + t for t = 1, . . . , n.

where t = t - 1 + ut

y t-1 = b1 + b2Xt -1, 2 + b3 Xt -1, 3 + t -1

Lag once and multiply by

Subtract from the original and move y t-1 to right side:

yt = b1-+ b2Xt 2 - Xt-1, 2 + b3(Xt 3 Xt-1, 3)+ y t-1+ ut

yt = b1-+ b2Xt 2 - Xt-1, 2 + b3(Xt 3 -Xt-1, 3) + y t-1+ ut

Now Durbin separates out the terms as follows:

yt = b1-+ b2Xt 2 - b2Xt-1 2 + b3Xt 3 - b3Xt-1 3+ y t-1+ ut

The structural (restricted,behavorial) equation is:

The corresponding reduced form (unrestricted) equation is:

yt = 1+ 2Xt, 2 + 3Xt-1, 2 + 4Xt, 3 + 5Xt-1, 3 + 6yt-1+ u t

1 = b1- 2 = b23= - b2 4 = b3 5= - b3 6=

Given OLS estimates: 1 2 3 4 5 6

^^^^^ ^

we can get three separate and distinct estimates for

^^ 6^^

These three separate estimates of are in conflict !!!It is difficult to know which one to use as “the”legitimate estimate of Durbin used the last one.

1 = b1- 2 = b23= - b2 4 = b3 5= - b3 6=

The problem with Durbin’s Method is that it ignoresthe inherent nonlinear restrictions implied by this structural model. To get a single (i.e. unique) estimatefor the implied nonlinear restrictions must be incorporated directly into the estimation process.

Consequently, the above structural equation should beestimated using a nonlinear method such as theGauss-Newton algorithm for nonlinear least squares.

yt = b1-+ b2Xt 2 - b2Xt -1, 2 + b3Xt 3 - b3Xt -1, 3+ yt-1+ ut

yt = b1-+ b2Xt 2 - b2Xt-1, 2 + b3Xt 3 - b3Xt-1, 3+ yt-1+ ut

f’(Xt,b) [ ]yt

= =X t, 2 X t-1,2)

=X t, 3 X t-1,3)

= ( - b1-b2Xt-1,2 - b3Xt-1,3+ y t-1 )

where yt(m)

yt - f(Xt,bm) + f’(Xt,b(m) b(m؛

(m+1) = [ f’(X,bm)T f’(X,bm)]-1 f’(X,b(m)

f(Xt,b) = b1-+ b2Xt 2 - b2Xt-1 2 + b3Xt 3 - b3Xt-1 3+ y t-1

b(m) =

Iterate until convergence.

f’(Xt,bm [ ]yt

Distributed

Lag Models

Chapter 15

The Distributed Lag Effect

Economic actionat time t

Effect at time t

Effect at time t+1

Effect at time t+2

Unstructured Lags

yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et

“n” unstructured lags

no systematic structure imposed on the ’s

the ’s are unrestricted

Problems with Unstructured Lags

1. n observations are lost with n-lag setup.

2. high degree of multicollinearity among xt-j’s.

3. many degrees of freedom used for large n.

4. could get greater precision using structure.

The Arithmetic Lag Structure

proposed by Irving Fisher (1937)

the lag weights decline linearly

Imposing the relationship:

# = (n - # + 1)

0 = (n+1) 1 = n 2 = (n-1) 3 = (n-2) n-2 = 3 n-1 = 2 n = only need to estimate one coefficient, ,

instead of n+1 coefficients, 0 , ... , n .

Arithmetic Lag Structure

yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et

yt = + (n+1) xt + n xt-1 + (n-1) xt-2 + . . . + xt-n + et

Step 1: impose the restriction: # = (n - # + 1)

Step 2: factor out the unknown coefficient, .

yt = + [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et

Step 3: Define zt .

yt = + [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et

zt = [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n]

Step 5: Run least squares regression on:

yt = + zt + et

Step 4: Decide number of lags, n.

For n = 4: zt = [ 5xt + 4xt-1 + 3xt-2 + 2xt-3 + xt-4]

0 = (n+1)

2 = (n-1)

0 1 2 . . . . . n n+1

linear lag structure

Polynomial Lag Structureproposed by Shirley Almon (1965)

the lag weights fit a polynomial

where i = 1, . . . , n p = 2 and n = 4

For example, a quadratic polynomial:0 = 0

1 = 0 + 1 + 2

2 = 0 + 21 + 42

3 = 0 + 31 + 92

4 = 0 + 41 + 162

n = the length of the lagp = degree of polynomial

where i = 1, . . . , ni = 0 + 1i + 2i +...+ pi 2 p

i = 0 + 1i + 2i 2

Polynomial Lag Structureyt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 +4 xt-4 + et

yt = + 0xt + 0 + 1 + 2xt-1 + (0 + 21 + 42)xt-2

+ (0 + 31 + 92)xt-3+ (0 + 41 + 162)xt-4 + et

Step 2: factor out the unknown coefficients: 0, 1, 2.

yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4] + 1 [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt-4] + 2 [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et

Step 1: impose the restriction: i = 0 + 1i + 2i 2

Polynomial Lag Structure

Step 3: Define zt0 , zt1 and zt2 for 0 , 1 , and 2.

yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4] + 1 [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt-4] + 2 [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et

z t0 = [xt + xt-1 + xt-2 + xt-3 + xt-4]

z t1 = [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt- 4 ]

z t2 = [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt- 4]

Step 4: Regress yt on zt0 , zt1 and zt2 .

yt = + 0 z t0 + 1 z t1 + 2 z t2 + et

Step 5: Express i‘s in terms of 0 , 1 , and 2.

^ ^ ^ ^

1 = 0 + 1 + 2

2 = 0 + 21 + 42

3 = 0 + 31 + 92

4 = 0 + 41 + 162

.. . .

0 1 2 3 4 i

Figure 15.3

Geometric Lag Structure

yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + et

infinite distributed lag model:

yt = + i xt-i + eti = 0

(15.3.1)

geometric lag structure:

i = i where || < 1 and i

yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + . . . + et

yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et

infinite unstructured lag:

infinite geometric lag:

Substitute i = i

0 = 1 = 2 = 2

interim multiplier (3-period) :

impact multiplier :

long-run multiplier :

yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et

+ + + + . . .

Figure 15.5

. .0 1 2 3 4 i

geometrically declining weights

Problem:

How to estimate the infinite number of geometric lag coefficients ???

yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et

Answer:

Use the Koyck transformation.

The Koyck Transformation

yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et

yt yt-1 = + xt + (et et-1)

Lag everything once, multiply by and subtract from original:

yt-1 = + xt-1 + xt-2 + xt-3 + . . .) + et-1

The Koyck Transformation

yt yt-1 = + xt + (et et-1)

yt = + yt-1 + xt + (et et-1)

Solve for yt by adding yt-1 to both sides:

yt = + yt-1 + xt + t

The Koyck Transformationyt = + yt-1 + xt + (et et-1)

yt = + yt-1 + xt + t

Defining = , = , and = ,

use ordinary least squares:

= ^ ^ ^

The original structuralparameters can now beestimated in terms ofthese reduced formparameter estimates.

yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et^ ^ ^ ^ ^ ^

yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + . . . + et ^^ ^ ^ ^ ^

Durbin’s h-test for autocorrelation

T 11 ( T 1)[se(b2)]2

h = 1 d2

h = Durbin’s h-test statistic

d = Durbin-Watson test statistic

se(b2) = standard error of the estimate b2

T = sample size

Estimates inconsistent if geometric lag model is autocorrelated,but Durbin-Watson test is biased in favor of no autocorrelation.

yt = + x*t + et

Adaptive Expectations

yt = credit card debt

x*t = expected (anticipated) income

(x*t is not observable)

x*t - x*t-1 = (xt-1 - x*t-1)

adjust expectations based on past realization:

x*t - x*t-1 = (xt-1 - x*t-1)

x*t = xt-1 + (1- ) x*t-1

rearrange to get:

xt-1 = [x*t - (1- ) x*t-1]

yt = + x*t + et

Lag this model once and multiply by (1):

yt = - (1)yt-1+ [x*t - (1)x*t-1]

+ et - (1)et-1

subtract this from the original to get:

(1)yt-1 = (1) + (1) x*t-1 + (1)et-1

yt = - (1)yt-1+ [x*t - (1)x*t-1]

+ et - (1)et-1

Since xt-1 = [x*t - (1- ) x*t-1] we get:

yt = - (1)yt-1+ xt-1 + ut

where ut = et - (1)et-1

yt = - (1)yt-1+ xt-1 + ut

yt = 1 + 2yt-1+ 3xt-1 + ut

Use ordinary least squares regression on:

and we get:

^ = (12)

^ ^ =(12)

Partial Adjustment

yt - yt-1 = (y*t - yt-1)

inventories partially adjust , 0 < < 1,

towards optimal or desired level, y*t :

y*t = + xt + et

yt - yt-1 = (y*t - yt-1)

= ( + xt + et - yt-1)

= + xt - yt-1+ et

yt = + (1 - yt-1 + xt + et

Solving for yt :

yt = + (1 - yt-1 + xt + et

yt = 1 + 2yt-1+ 3xt + t

^ = (12) ^ ^

Use ordinary least squares regression to get:

Time Series

Analysis

Chapter 16

Previous Chapters used Economic Models

1. economic model for dependent variable of interest.

2. statistical model consistent with the data.

3. estimation procedure for parameters using the data.

4. forecast variable of interest using estimated model.

Times Series Analysis does not use this approach.

Time Series Analysis is useful for short term forecasting only.

Time Series Analysis does not generallyincorporate all of the economic relationships found in economic models.

Times Series Analysis uses more statistics and less economics.

Long term forecasting requires incorporating more involvedbehavioral economic relationships into the analysis.

Univariate Time Series Analysis can be used to relate the current values of a single economicvariable to:

1. its past values

2. the values of current and past random errors

Other variables are not used in univariate time series analysis.

1. autoregressive (AR)

2. moving average (MA)

3. autoregressive moving average (ARMA)

Three types of Univariate Time Series Analysisprocesses will be discussed in this chapter:

1. its past values.

2. the past values of the other forecasted variables.

3. the values of current and past random errors.

Multivariate Time Series Analysis can be used to relate the current value of each of several economic variables to:

Vector autoregressive models discussed later inthis chapter are multivariate time series models.

First-Order Autoregressive Processes, AR(1):

yt = + 1yt-1+ et, t = 1, 2,...,T. (16.1.1)

is the intercept.

1 is parameter generally between -1 and +1.

et is an uncorrelated random error with mean zero and variance e

Autoregressive Process of order p, AR(p) :

yt = + 1yt-1 + 2yt-2 +...+ pyt-p + et (16.1.2)

is the intercept.

i’s are parameters generally between -1 and +1.

AR models always have one or more lagged dependent variables on the right hand side.

Consequently, least squares is no longer abest linear unbiased estimator (BLUE), but it does have some good asymptotic properties including consistency.

Properties of least squares estimator:

AR(2) model of U.S. unemployment rates

yt = 0.5051 + 1.5537 yt-1 - 0.6515 yt-2

(0.1267) (0.0707) (0.0708)

Note: Q1-1948 through Q1-1978 from J.D.Cryer (1986) see unempl.dat

positive

negative

Choosing the lag length, p, for AR(p):

The Partial Autocorrelation Function (PAF)

The PAF is the sequence of correlations between(yt and yt-1), (yt and yt-2), (yt and yt-3), and so on,given that the effects of earlier lags on yt are held constant.

yt = 0.5 yt-1 + 0.3 yt-2 + et

02 / T

kk is the last (kth) coefficient in a kth order AR process.

This sample PAF suggests a second order process AR(2) which is correct.

Data simulatedfrom this model:

Using AR Model for Forecasting:

unemployment rate: yT-1 = 6.63 and yT = 6.20

yT+1 = + 1 yT + 2 yT-1

= 0.5051 + (1.5537)(6.2) - (0.6515)(6.63)

= 5.8186

^ ^ ^ ^

yT+2 = + 1 yT+1 + 2 yT

= 0.5051 + (1.5537)(5.8186) - (0.6515)(6.2)

= 5.5062

^ ^ ^ ^

yT+1 = + 1 yT + 2 yT-1

= 0.5051 + (1.5537)(5.5062) - (0.6515)(5.8186)

= 5.2693

^ ^ ^ ^

Moving Average Process of order q, MA(q):

yt = + et + 1et-1 + 2et-2 +...+ qet-q + et (16.2.1)

is the intercept.

i‘s are unknown parameters.

An MA(1) process:

yt = + et + 1et-1 (16.2.2)

Minimize sum of least squares deviations:

S(,1) = et = yt - -1et-1) (16.2.3)2

stationary:A stationary time series is one whose mean, variance,and autocorrelation function do not change over time.

nonstationary:A nonstationary time series is one whose mean,variance or autocorrelation function change over time.

Stationary vs. Nonstationary 16.16

yt = z t - z t-1

First Differencing is often used to transforma nonstationary series into a stationary series:

where z t is the original nonstationary series

and yt is the new stationary series.

Choosing the lag length, q, for MA(q):

The Autocorrelation Function (AF)

The AF is the sequence of correlations between(yt and yt-1), (yt and yt-2), (yt and yt-3), and so on,without holding the effects of earlier lags on yt constant.

The PAF controlled for the effects of previous lagsbut the AF does not control for such effects.

yt = et 0.9 et-1

02 / T

rkk is the last (kth) coefficient in a kth order MA process.

This sample AF suggests a first order process MA(1) which is correct.

Data simulatedfrom this model:

Autoregressive Moving AverageARMA(p,q)

An ARMA(1,2) has one autoregressive lagand two moving average lags:

yt = + 1yt-1 + et + 1et-1 + 2 et-2

Integrated Processes

A time series with an upward or downwardtrend over time is nonstationary.

Many nonstationary time series can be made stationary by differencing them one or more times.

Such time series are called integrated processes.

The number of times a series must be differenced to make it stationary is theorder of the integrated process, d.

An autocorrelation function, AF, with large, significant autocorrelationsfor many lags may require more thanone differencing to become stationary.

Check the new AF after each differencingto determine if further differencing is needed.

Unit Root

zt = 1zt -1 + + et + 1et -1 (16.3.2)

-1 < 1 < 1 stationary ARMA(1,1)

1 = 1 nonstationary process

1 = 1 is called a unit root

Unit Root Tests

zt = 1zt -1 + + et + 1et -1 (16.3.3)

Testing1 = 0 is equivalent to testing 1 = 1

zt - zt -1 = (1- 1)zt -1 + + et + 1et -1

where zt = zt - zt -1 and 1 = 1- 1 *

Unit Root Tests

H0:1 = 0 vs. H1:1 < 0 (16.3.4)**

Computer programs typically use one of the following tests for unit roots:

Dickey-Fuller Test

Phillips-Perron Test

Autoregressive Integrated Moving Average ARIMA(p,d,q)

An ARIMA(p,d,q) model represents an AR(p) - MA(q) process that has been differenced (integrated, I(d)) d times.

yt = + 1yt-1 +...+ pyt-p + et + 1et-1 +... + q et-q

The Box-Jenkins approach:

1. Identificationdetermining the values of p, d, and q.

2. Estimationlinear or nonlinear least squares.

3. Diagnostic Checkingmodel fits well with no autocorrelation?

4. Forecastingshort-term forecasts of future yt values.

Vector Autoregressive (VAR) Models

yt = 0+ 1yt-1 +...+ pyt-p + 1xt-1 +... + p xt-p + et

xt = 0+ 1yt-1 +...+ pyt-p + 1xt-1 +... + p xt-p + ut

Use VAR for two or more interrelated time series:

1. extension of AR model.

2. all variables endogenous.

3. no structural (behavioral) economic model.

4. all variables jointly determined (over time).

5. no simultaneous equations (same time).

Vector Autoregressive (VAR) Models16.29

The random error terms in a VAR modelmay be correlated if they are affected byrelevant factors that are not in the modelsuch as government actions or national/international events, etc.

Since VAR equations all have exactly the same set of explanatory variables, the usualseemingly unrelation regression estimationproduces exactly the same estimates asleast squares on each equation separately.

Consequently, regardless of whetherthe VAR random error terms arecorrelated or not, least squares estimationof each equation separately will provideconsistent regression coefficient estimates.

Least Squares is Consistent

VAR Model Specification

To determine length of the lag, p, use:

2. Schwarz’s SIC criterion

1. Akaike’s AIC criterion

These methods were discussed in Chapter 15.

Spurious Regressions

yt = 1+ 2 xt + t

where t = 1 t-1 + t

-1 <1 < 1 I(0) (i.e. d=0)

1 = 1 I(1) (i.e. d=1)

If 1 =1 least squares estimates of 2 mayappear highly significant even when true 2 = 0 .

Cointegration

yt = 1+ 2 xt + t

If xt and yt are nonstationary I(1)

we might expect that t is also I(1).

However, if xt and yt are nonstationary I(1)

but t is stationary I(0), then xt and yt are

said to be cointegrated.

Cointegrated VAR(1) Model

yt = 0+ 1yt-1 + 1xt-1 + et

xt = 0+ 1yt-1 + 1xt-1 + ut

VAR(1) model:

If xt and yt are both I(1) and are cointegrated,

use an Error Correction Model, instead of VAR(1).

Error Correction Model

yt = 0+ (1-1)yt-1 + 1xt-1 + et

xt = 0+ 1yt-1 + (1-1)xt-1 + ut

yt = yt - yt-1 and xt = xt - xt-1

(continued)

yt = 0+ 1(yt-1 - 1- 2 xt-1) + et *

xt = 0+ 2(yt-1 - 1- 2 xt-1) + ut *

0= 0 + 11

0= 0 + 21* 2= 1

1 - 12=1

yt-1 = 1+ 2 xt-1 + t-1

Estimate by least squares:

to get the residuals:

t-1 = yt-1 - 1- 2 xt-1 ^ ^ ^

Estimating an Error Correction Model

Step 1:Step 1:

Estimate by least squares:

Estimating an Error Correction Model

Step 2:Step 2:

yt = 0+ 1 t-1 + et *

xt = 0+ 2 t-1 + ut *

Using cointegrated I(1) variables in aVAR model expressed solely in termsof first differences and lags of firstdifferences is a misspecification.

The correct specification is to use an

Chapter 17

Guidelines forResearch Project

��

Formulation economic ====> econometric.

Estimation selecting appropriate method.

Interpretation how the xt’s impact on the yt .

Inference testing, intervals, prediction.

What Book Has Covered17.2

Topics for This Chapter1. Types of Data by Source

2. Nonexperimental Data

3. Text Data vs. Electronic Data

4. Selecting a Topic

5. Writing an Abstract

6. Research Report Format

Types of Data by Source

i) Experimental Data from controlled experiments.

ii) Observational Datapassively generated by society.

iii) Survey Datadata collected through interviews.

Time vs. Cross-Section

Time Series Datadata collected at distinct points in time(e.g. weekly sales, daily stock price, annual budget deficit, monthly unemployment.)

Cross Section Datadata collected over samples of units, individuals, households, firms at a particular point in time. (e.g. salary, race, gender, unemployment by state.)

Micro vs. Macro

Micro Data:data collected on individual economicdecision making units such as individuals,households or firms.

Macro Data:data resulting from a pooling or aggregatingover individuals, households or firms at thelocal, state or national levels.

Flow vs. Stock

Flow Data:outcome measured over a period of time,such as the consumption of gasoline duringthe last quarter of 1997.

Stock Data:outcome measured at a particular point intime, such as crude oil held by Chevron inUS storage tanks on April 1, 1997.

Quantitative vs. Qualitative

Quantitative Data:outcomes such as prices or income that maybe expressed as numbers or some transfor-mation of them (e.g. wages, trade deficit).

Qualitative Data:outcomes that are of an “either-or” nature(e.g. male, home owner, Methodist, bought car last year, voted in last election).

International Data

International Financial Statistics (IMF monthly).

Basic Statistics of the Community (OECD annual).

Consumer Price Indices in the European Community (OECD annual).

World Statistics (UN annual).

Yearbook of National Accounts Statistics (UN).

FAO Trade Yearbook (annual).

United States Data

Survey of Current Business (BEA monthly).

Handbook of Basic Economic Statistics (BES).

Monthly Labor Review (BLS monthly).

Federal Researve Bulletin (FRB monthly).

Statistical Abstract of the US (BC annual).

Economic Report of the President (CEA annual).

Economic Indicators (CEA monthly).

Agricultural Statistics (USDA annual).

Agricultural Situation Reports (USDA monthly).

State and Local Data

State and Metropolitan Area Data Book (Commerce and BC, annual).

CPI Detailed Report (BLS, annual).

Census of Population and Housing (Commerce, BC, annual).

County and City Data Book (Commerce, BC, annual).

Citibase on CD-ROM

• Financial series: interest rates, stock market, etc.

• Business formation, investment and consumers.

• Construction of housing.

• Manufacturing, business cycles, foreign trade.

• Prices: producer and consumer price indexes.

• Industrial production.

• Capacity and productivity.

• Population.

Citibase on CD-ROM(continued)

• Labor statistics: unemployment, households.

• National income and product accounts in detail.

• Forecasts and projections.

• Business cycle indicators.

• Energy consumption, petroleum production, etc.

• International data series including trade statistics.

Resources for Economists

Resources for Economists by Bill Goffe

http://econwpa.wustl.edu/EconFAQ/EconFAQ.html

Bill Goffe provides a vast database of information about the economics profession including economic organizations, working papers and reports, and economic data series.

Internet Data Sources

• Shortcut to All Resources.

• Macro and Regional Data.

• Other U.S. Data.

• World and Non-U.S. Data.

• Finance and Financial Markets.

• Data Archives.

• Journal Data and Program Archives.

A few of the items on Bill Goffe’s Table of Contents:

Useful Internet Addresses

http://seamonkey.ed.asu.edu/~behrens/teach/WWW_data.html

http://www.sims.berkeley.edu/~hal/pages/interesting.html

http://www.stls.frb.org FED RESERVE BK - ST. LOUIS

http://www.bls.gov BUREAU OF LABOR STATISTICS

http://nber.harvard.edu NAT’L BUR. ECON. RESEARCH

http://www.inform.umd.edu:8080/EdRes/Topic/EconData/.www/econdata.html UNIVERSITY OF MARYLAND

http://www.bog.frb.fed.us FEB BOARD OF GOVERNORS

http://www.webcom.com/~yardeni/economic.html

i) identify the population of interest.

ii) designing and selecting the sample.

iii) collecting the information.

iv) data reduction, estimation and inference.

The survey process has four distinct aspects:

Controlled Experiments

1. Labor force participation: negative income tax:guaranteed minimum income experiment.

2. National cash housing allowance experiment:impact on demand and supply of housing.

3. Health insurance: medical cost reduction: sensitivity of income groups to price change.

4. Peak-load pricing and electricity use:daily use pattern of residential customers.

Controlled experiments were done on these topics:

Economic Data Problems

I. poor implicit experimental design

(i) collinear explanatory variables.

(ii) measurement errors.

II. inconsistent with theory specification

(i) wrong level of aggregation.

(ii) missing observations or variables.

(iii) unobserved heterogeneity.

Selecting a Topic

• “What am I interested in?”

• Well-defined, relatively simple topic.

• Ask prof for ideas and references.

• Journal of Economic Literature (ECONLIT)

• Make sure appropriate data are available.

• Avoid extremely difficult econometrics.

• Plan your work and work your plan.

General tips for selecting a research topic:

��

Writing an Abstract

(i) concise statement of the problem.

(ii) key references to available information.

(iii) description of research design including:(a) economic model(b) statistical model(c) data sources(d) estimation, testing and prediction

(iv) contribution of the work

Abstract of less than 500 words should include:

Research Report Format

1. Statement of the Problem. 2. Review of the Literature. 3. The Economic Model. 4. The Statistical Model. 5. The Data. 6. Estimation and Inferences Procedures. 7. Empirical Results and Conclusions. 8. Possible Extensions and Limitations. 9. Acknowledgments.10. References.

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