housing starts forecast

John Montgomery

Econ 401/Dr. TownsendDecember 7, 2009

Appendix 14.1 is a highly aggregated model of real gross domestic product and its

major components. The Model contains 11 behavioral equations and two identities. One

of these identities is for real disposable income, and the other is the accounting identity

for real GDP. Each equation within the model is estimated using two stage least squares.

There are 12 endogenous variables: personal consumption expenditures, GDP, rate of

growth of CPI, nonresidential fixed investment, change in business inventories,

residential fixed investement, imports of goods and services, average yield on AAA

corporate bonds, interest rate on 3-month treasury bills, personal and indirect business tax

payments, civilian unemployment rate, wage inflation, and disposable personal income.

In addition to these endogenous variables, there are 9 exogenous variables: government

purchases of goods and services, potential GDP, money stock, household net worth, rate

of growth of oil prices, corporate profits, rate of growth of labor productivity, transfer

payments to persons, and exports of goods and services.

The instruments used for the individual behavioral equations differ compared to

what we will be using for our model. Furthermore this model uses two-stage least

squares for each of the equations, and we use ordinary least squares for the recursive

equations.

Comparatively the model provides a good forecast, and the flow chart is a good

representation of the equation visually.

Case set four calls for us to create a simplified structural model of the U.S.

economy. The model uses the Fair method, which uses two stage least squares, and

includes the lagged dependent and independent variables as instruments. These lagged

variables are included as such in order to obtain consistent parameter estimates when

autocorrelated disturbances create a problem.

The model contains 11 behavioral equations, and two identities. The majority of

the equations are estimated using two stage least squares, although there are three

recursive equations which are estimated using the ordinary least squares method. Using

quarterly data from 1960-1993 I have created a historical simulation which I will explain

here.

Dependent Variable: TAXMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:36Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 7 iterationsInstrument list: C GDPPOT INFL INR INV IR M M2 RL RS X YPD GDP(-1) TAX(-1)

Lagged dependent variable & regressors

added to instrument list

Variable Coefficient Std. Error t-Statistic Prob.

C -3.967408 22.99851 -0.172507 0.8633GDP 0.186861 0.005053 36.98054 0.0000AR(1) 0.781575 0.054284 14.39795 0.0000

R-squared 0.995351 Mean dependent var 790.8930Adjusted R-squared 0.995281 S.D. dependent var 235.3508S.E. of regression 16.16703 Sum squared resid 34762.61F-statistic 14231.47 Durbin-Watson stat 2.331248Prob(F-statistic) 0.000000

Inverted AR Roots .78

The first equation examined is the equation for tax. It is a very simple equation, and is

the calculation of total business and personal taxes. Its instruments are potential gdp, inflation,

nonresidential fixed investment, change in business inventories, residential fixed investment,

imports of goods and services, the money stock, average yield on AAA corporate bonds, interest

rates on three-month treasury bills, exports, disposable personal income, gross domestic product

lagged by one quarter, and finally itself lagged by one quarter. The high r-squared number

indicates that we should have a very good fitting line, and we also see a Durbin-Watson statistic

within the acceptable range. I have used the auto-regressive model to help correct for any serial

correlation, so that explains why we have such a good D-W stat.

200

400

600

800

1000

1200

1400

1960 1965 1970 1975 1980 1985 1990

TAX TAX (Baseline)

Above is the historical simulation of taxes, and as our r-squared value had indicated we

have a decently nice fitting line. The MAPE for the historical simulation is .05%.

1200

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1280

1300

1320

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

TAX TAX (Scenario 1)

Above is the ex-post ante forecast for the tax equation. We have been able to generate a

fairly strong forecast which has a MAPE of .019.

Dependent Variable: CONSMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:38Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 44 iterationsInstrument list: C G GDPPOT INFL INR INV IR M M2 RL WINF X CONS CONS(-2) NETWRTH(-1) YPD(-1)




C -146.5984 35.31959 -4.150627 0.0001YPD 0.192170 0.039694 4.841345 0.0000

NETWRTH 0.040520 0.009706 4.174715 0.0001RS -5.241978 1.330045 -3.941204 0.0001

CONS(-1) 0.586638 0.085641 6.849938 0.0000AR(1) 0.406659 0.116241 3.498412 0.0006

R-squared 0.999569 Mean dependent var 2834.458Adjusted R-squared 0.999552 S.D. dependent var 873.7046

S.E. of regression 18.49246 Sum squared resid 44456.23F-statistic 60252.24 Durbin-Watson stat 2.165461Prob(F-statistic) 0.000000


The above table is the results of the two stage least squares regression for the

consumption equation. Personal consumption represents two-thirds of GDP and is one of the

most important behavioral equations within the entire model. Because of the presence of the

lagged dependent variable in the equation, and in accordance with Fair’s method, I have included

the consumption variable lagged twice upon itself in the instruments. In addition to this I have

included a lagged variable of both net worth and personal disposable income because they are

also endogenous variables. Again, we notice a high r-squared value, indicating a good-fitting

line. Also, the Durbin-Watson statistic is within its accepted values, which has happened again

because of the addition of the autoregressive model. The negative coefficient present for the

variable representing the three-month treasury bill interest rates makes sense as one can

assume that as consumption increases, the interest on these would in turn decrease. The

positive coefficients for both net worth and disposable personal income also makes sense as it is

only logical to assume that consumption would increase as these two variables do as well.

1000

1500

2000

2500

3000

3500

4000

4500

1960 1965 1970 1975 1980 1985 1990

CONS CONS (Baseline)

Above is the graph for the historical simulation of consumption, and as our r-

squared value indicates we have a strong fit; the MAPE for the historical simulation of

consumption is .017%.

4400

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4520

4560

4600

4640

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

CONS CONS (Scenario 1)

Above is a graphical representation of the ex-post ante forecast for the

consumption equation. Although it looks like it is dipping far below the actual line, it

really isn’t, as can be seen in a graphical representation including the historical

simulation.

1200

1600

2000

2400

2800

3200

3600

4000

4400

4800

1965 1970 1975 1980 1985 1990 1995

CONS (Scenario 1)CONSCONS (Baseline)

As you can see there is actually a very close fitting ex-post forecast provided, and

the MAPE of .01%.

Dependent Variable: MMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 21:14Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 5 iterationsInstrument list: C CONS G GDP GDPPOT INFL INR INV M2 RL RS X YPD(-1)




M(-1) 0.997952 0.019728 50.58575 0.0000C -5.780378 8.064005 -0.716812 0.4748

YPD 0.002797 0.003503 0.798507 0.4260AR(1) 0.120054 0.089127 1.347008 0.1803



The next equation is for imports of goods and services. The r-squared value is strong,

and the Durbin-Watson statistic is again within the acceptable region. The positive coefficient of

personal disposable income makes sense in the fact that the more money people have, the more

they will spend, and the more goods and services we will import.

100

200

300

400

500

600

700

800

1960 1965 1970 1975 1980 1985 1990

M M (Baseline)

The historical simulation shows a decent fitting line, and the simulation has become a

strong trend. The MAPE for the import equation is .092%.

780

800

820

840

860

880

900

920

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

M (Scenario 1) M

100

200

300

400

500

600

700

800

900

1000

1965 1970 1975 1980 1985 1990 1995

M (Scenario 1) M M (Baseline)

The first graph above shows the ex-post forecast, and the graph directly below shows the

ex-post forecast included with the actual numbers, and the historical simulation. The MAPE for

the ex-post forecast is .06%, and it continues along the trend that the historical simulation begins.

Dependent Variable: INRMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:42Sample (adjusted): 1960Q2 1993Q4Included observations: 135 after adjustmentsConvergence achieved after 26 iterationsInstrument list: C CONS G GDPPOT INFL INV IR M M2 X YPD GDP( -1) INR(-1) RL(-5)




C 21.67766 108.1270 0.200483 0.8414GDP 0.107208 0.015837 6.769528 0.0000

RL(-4) -6.854006 3.604487 -1.901520 0.0594AR(1) 0.977314 0.021144 46.22132 0.0000



Moving forward we next look at the equation for nonresidential investment, and

immediately we notice that it has a positive effect on aggregate economic activity. However, it has

a negative effect on the opportunity cost of investment. Again, we see a high r-squared value,

which translates to a good fitting line.

100

200

300

400

500

600

700

800

1960 1965 1970 1975 1980 1985 1990

INR INR (Baseline)

The historical simulation shows a line that doesn’t fit quite as well as many of the

previous equations historical simulations have, and we see a MAPE of .144%.

620

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740

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

INR (Scenario 1) INR

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500

600

700

800

1960 1965 1970 1975 1980 1985 1990

INR (Scenario 1)INRINR (Baseline)

As we look at the above graphs we also see a larger separation between the actual

numbers, and the ex-post forecast. The MAPE for nonresidential investment is .058%.

Dependent Variable: IRMethod: Least SquaresDate: 12/07/09 Time: 20:43

Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 33 iterations


C 12.99230 60.40403 0.215090 0.8300YPD(-1) 0.048791 0.013085 3.728651 0.0003RS(-1) -3.810494 0.941601 -4.046825 0.0001AR(1) 0.949368 0.029951 31.69789 0.0000

R-squared 0.961015 Mean dependent var 190.7934Adjusted R-squared 0.960129 S.D. dependent var 43.58628S.E. of regression 8.703175 Akaike info criterion 7.194223Sum squared resid 9998.374 Schwarz criterion 7.279890Log likelihood -485.2072 F-statistic 1084.643Durbin-Watson stat 1.109263 Prob(F-statistic) 0.000000


Residential investment is a variable that reflects household demand for new homes. It is

estimated as a function of real disposable income and the cost of borrowing. We are using the

interest rates for three-month treasury bills as a proxy for mortgage rates.

80

120

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200

240

280

1960 1965 1970 1975 1980 1985 1990

IR IR (Baseline)

The historic simulation shows an actual set of values that oscillates regularly between

peaks and troughs, but the simulation almost begins to show a trend. The MAPE for the historical

simulation is .13%.

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260

264

268

272

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

IR IR (Scenario 1)

The ex-post forecast shows a forecast that falls below the values of the actual numbers.

The MAPE is .032%.

Dependent Variable: INVMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 21:23Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 10 iterationsInstrument list: C CONS G GDPPOT INFL INR IR M M2 RL RS X INV( -2) (GDP-CONS-GDP(1)+CONS(-1))




C 2.837186 1.626528 1.744321 0.0834D(GDP-CONS) 0.360108 0.058365 6.169931 0.0000

INV(-1) 0.709656 0.054278 13.07454 0.0000AR(1) -0.182547 0.106412 -1.715472 0.0886

R-squared 0.675370 Mean dependent var 21.58603

Adjusted R-squared 0.667992 S.D. dependent var 22.24099S.E. of regression 12.81529 Sum squared resid 21678.58F-statistic 50.06773 Durbin-Watson stat 2.073371Prob(F-statistic) 0.000000

Inverted AR Roots -.18

The next equation is for the change in business inventories. Reasearch has shown that

much of the variation in real output growth over the course of a business cycle can be attributed

to variations in the rate of inventory accumulation. This equation is estimated as a function of the

change in the difference between total output and consumption.

-80

-40

0

40

80

120

160

200

1960 1965 1970 1975 1980 1985 1990

INV INV (Baseline)

The historic simulation of business inventories is represented graphically above.

Immediately one’s eyes would be drawn to the beginning of the cycle in which there is an

impossibly large peak in the simulation. This peak could be controlled through the use of a

dummy variable, but doesn’t affect the simulation greatly. The MAPE of the historical simulation

is the largest of all the equations at 2..77%. However, it is important to note that this number is

still below the 5% threshold that is generally considered in good form for a forecast.

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94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

INV (Scenario 1) INV

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0

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160

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1960 1965 1970 1975 1980 1985 1990 1995

INV (Scenario 1)INVINV (Baseline)

The above graphs show the ex-post forecast for the equation regarding business

inventories. The MAPE improves from the historical simulation to .449%.

Dependent Variable: RSMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:52

Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 8 iterationsInstrument list: C CONS G INR INV IR M RL X INFL(-1) RS(-1) M2(-1) YPD(-1)




C -44.76644 12.53438 -3.571492 0.0005YPD 0.014637 0.003054 4.792746 0.0000M2 -0.021874 0.005354 -4.085905 0.0001

INFL 0.303852 0.129569 2.345099 0.0205AR(1) 0.956617 0.022165 43.15966 0.0000



Short-term interest rates (rates on three-month treasury bills) are modeled as a

normalization of a traditional money demand equation. When personal disposable income

increasing demand for money increases, but decreases when real short-term interest rates rise

as the opportunity cost of holding money increases. The r-squared values for this equation are

lower than other equations, and that makes sense. Interest rates are more volatile than any of

the other variables, and therefore much more difficult to predict.

-4

0

4

8

12

16

20

1960 1965 1970 1975 1980 1985 1990

RS RS (Baseline)

As you can see the historical simulation isn’t quite as fitted as many of the other

simulations that I have introduced today. The spike in the 80’s is consistent with Paul Volker

increasing the interest rates to battle inflation. The MAPE for this historical simulation is .59%.

3.2

3.6

4.0

4.4

4.8

5.2

5.6

6.0

6.4

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

RS RS (Scenario 1)

The MAPE for the ex-post forecast is .179%.

Dependent Variable: RL

Method: Least SquaresDate: 12/07/09 Time: 20:53Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 9 iterations


C 0.301862 0.110459 2.732789 0.0071RS 0.188788 0.020330 9.286057 0.0000

RL(-1) 0.822268 0.021359 38.49828 0.0000AR(1) 0.213139 0.088868 2.398388 0.0179

R-squared 0.987126 Mean dependent var 8.211863Adjusted R-squared 0.986833 S.D. dependent var 2.743314S.E. of regression 0.314787 Akaike info criterion 0.555132Sum squared resid 13.08002 Schwarz criterion 0.640798Log likelihood -33.74896 F-statistic 3373.660Durbin-Watson stat 2.028879 Prob(F-statistic) 0.000000


This is the regression for average yield on AAA bonds. It is a member of the recursive

block, so it was run using only ordinary least squares.

0

4

8

12

16

20

1960 1965 1970 1975 1980 1985 1990

RL RL (Baseline)

The MAPE for the historic simulation is .38%.

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8.0

8.4

8.8

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

RL (Scenario 1) RL

The MAPE for the ex-post fore cast is .08%.

Dependent Variable: UR

Method: Least Squares

Date: 12/07/09 Time: 22:46

Sample (adjusted): 1960Q3 1993Q4

Included observations: 134 after adjustments

Convergence achieved after 8 iterations


C 6.582626 1.181766 5.570160 0.0000

(D(LOG(GDP)))-(D(LOG(GDPPOT))) -3.592488 2.730454 -1.315711 0.1906

AR(1) 0.973305 0.019730 49.33012 0.0000

R-squared 0.949410 Mean dependent var 6.178109

Adjusted R-squared 0.948637 S.D. dependent var 1.554937

S.E. of regression 0.352400 Akaike info criterion 0.774035

Sum squared resid 16.26835 Schwarz criterion 0.838912

Log likelihood -48.86033 F-statistic 1229.218

Durbin-Watson stat 0.650476 Prob(F-statistic) 0.000000


The unemployment rate is estimated according to a tradition Okun’s law equation relating

change in the unemployment rate to the change in GDP. It makes sense that there is a negative

effect of the unemployment rate on GDP. This equation is also in the recursive block, and

therefore is estimated using ordinary least squares.

3

4

5

6

7

8

9

10

11

1960 1965 1970 1975 1980 1985 1990

UR UR (Baseline)

The MAPE for the historical simulation is .19%.

5.4

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6.6

6.8

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

UR (Scenario 1) UR


Dependent Variable: WINFMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:55Sample: 1960Q1 1993Q4Included observations: 136Convergence achieved after 8 iterationsInstrument list: C CONS G GDP GDPPOT INFL(-1) INR INV IR M NETWRTH PRFT RL RS TR UR WINF(-1) X




C -14.26324 3.091834 -4.613198 0.0000INFL 0.691501 0.014116 48.98761 0.0000

UR(-2) 0.032879 0.096548 0.340545 0.7340PROD 0.152321 0.046329 3.287830 0.0013AR(1) 0.934047 0.033211 28.12424 0.0000



The annual rate of growth in wages will be a positive function of overall price inflation, a

negative function of the unemployment rate, and a positive function of productivity growth. We

have a very strong r-squared value, indicating a good fiiting line.

0

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100

120

1960 1965 1970 1975 1980 1985 1990

WINF WINF (Baseline)

The MAPE for the Historic simulation is .11%

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110

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

WINF WINF (Scenario 1)

The MAPE for the ex-post forecast is .01%

Dependent Variable: INFLMethod: Two-Stage Least SquaresDate: 12/07/09 Time: 20:57Sample (adjusted): 1960Q2 1993Q4Included observations: 135 after adjustmentsConvergence achieved after 19 iterationsInstrument list: C CONS CONS(-2) G GDP(-1) GDPPOT INV IR M NETWRTH PRFT RL RS TR WINF(-1) X YPD




C 2.645615 2.952583 0.896034 0.3719WINF 0.676700 0.140916 4.802149 0.0000

CONS(-1) 0.000505 0.001582 0.318843 0.7504POIL 0.092758 0.022974 4.037453 0.0001

INFL(-1) 0.479845 0.090355 5.310660 0.0000AR(1) 0.926117 0.041998 22.05126 0.0000



The annual rate of growth in the consumer price index is estimated to be a function of

wage inflation, consumer demand, and oil prices. We have a high r-squared value, and the

Durbin-Watson statistic falls within the accepted values.

20

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100

120

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160

1960 1965 1970 1975 1980 1985 1990

INFL INFL (Baseline)

The MAPE for the historic simulation is .10%.

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154

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

INFL INFL (Scenario 1)


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7000

1960 1965 1970 1975 1980 1985 1990

GDP GDP (Baseline)

After completing estimations of all the equations we can simulate the model as a

complete system. The above simulation is the historical look at GDP. It is a good fitting line, and

we are ultimately given a MAPE of .05%

2000

3000

4000

5000

6000

7000

1960 1965 1970 1975 1980 1985 1990 1995

GDP (Scenario 1)GDPGDP (Baseline)

Above is a graph of the historic simulation, actual numbers, and ex-post forecast

combined into one. From this view we see that the ex-post forecast looks pretty good. Below is a

closer look at the ex-post forecast.

6400

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6800

6900

94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

GDP GDP (Scenario 1)

The MAPE based on this simulation is .008%. This is a strong forecast for the gross

domestic product.

0

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2000

3000

4000

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6000

1960 1965 1970 1975 1980 1985 1990

YPD YDP_0

Looking at the results for the disposable personal income equation confirm our findings

for gross domestic product. The steady growth of personal disposable income is consistent with

the growth of gross domestic product. The MAPE of the historical simulation for personal

disposable income is .06%.

0

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2000

3000

4000

5000

6000

1960 1965 1970 1975 1980 1985 1990 1995

YDP_1 YPD YDP_0

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94Q1 94Q2 94Q3 94Q4 95Q1 95Q2 95Q3 95Q4

YPD YDP_1

The MAPE for the ex-post forecast of personal disposable income is .01%.

housing starts forecast

Technology

squares method

tax equation

squares date

squares reg

rsquared value

potential gdp

adjusted r

cons method