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Principles of Econometrics, 4th Edition
Page 1Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Chapter 12Regression with Time-Series
Data:Nonstationary Variables
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition
Page 2Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1 Stationary and Nonstationary Variables12.2 Spurious Regressions12.3 Unit Root Tests for Nonstationarity12.4 Cointegration12.5 Regression When There Is No Cointegration
Chapter Contents
Principles of Econometrics, 4th Edition
Page 3Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The aim is to describe how to estimate regression models involving nonstationary variables– The first step is to examine the time-series
concepts of stationarity (and nonstationarity) and how we distinguish between them.
– Cointegration is another important related concept that has a bearing on our choice of a regression model
Principles of Econometrics, 4th Edition
Page 4Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1
Stationary and Nonstationary Variables
Principles of Econometrics, 4th Edition
Page 5Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The change in a variable is an important concept
– The change in a variable yt, also known as its first difference, is given by Δyt = yt – yt-1.
• Δyt is the change in the value of the variable y from period t - 1 to period t
12.1Stationary and Nonstationary
Variables
Principles of Econometrics, 4th Edition
Page 6Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1Stationary and Nonstationary
VariablesFIGURE 12.1 U.S. economic time series
Principles of Econometrics, 4th Edition
Page 7Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1Stationary and Nonstationary
VariablesFIGURE 12.1 (Continued) U.S. economic time series
Principles of Econometrics, 4th Edition
Page 8Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Formally, a time series yt is stationary if its mean and variance are constant over time, and if the covariance between two values from the series depends only on the length of time separating the two values, and not on the actual times at which the variables are observed
12.1Stationary and Nonstationary
Variables
Principles of Econometrics, 4th Edition
Page 9Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
That is, the time series yt is stationary if for all values, and every time period, it is true that:
12.1Stationary and Nonstationary
Variables
2
μ (constant mean)
var σ (constant variance)
cov , cov , γ (covariance depends on , not )
t
t
t t s t t s s
E y
y
y y y y s t
Eq. 12.1a
Eq. 12.1b
Eq. 12.1c
Principles of Econometrics, 4th Edition
Page 10Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1Stationary and Nonstationary
VariablesTable 12.1 Sample Means of Time Series Shown in Figure 12.1
Principles of Econometrics, 4th Edition
Page 11Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Nonstationary series with nonconstant means are often described as not having the property of mean reversion– Stationary series have the property of mean
reversion
12.1Stationary and Nonstationary
Variables
Principles of Econometrics, 4th Edition
Page 12Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The econometric model generating a time-series variable yt is called a stochastic or random process
– A sample of observed yt values is called a particular realization of the stochastic process• It is one of many possible paths that the
stochastic process could have taken
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
Principles of Econometrics, 4th Edition
Page 13Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The autoregressive model of order one, the AR(1) model, is a useful univariate time series model for explaining the difference between stationary and nonstationary series:
– The errors vt are independent, with zero mean and constant variance , and may be normally distributed
– The errors are sometimes known as ‘‘shocks’’ or ‘‘innovations’’
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
1 , 1t t ty y v
2σv
Eq. 12.2a
Principles of Econometrics, 4th Edition
Page 14Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
FIGURE 12.2 Time-series models
Principles of Econometrics, 4th Edition
Page 15Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
FIGURE 12.2 (Continued) Time-series models
Principles of Econometrics, 4th Edition
Page 16Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The value ‘‘zero’’ is the constant mean of the series, and it can be determined by doing some algebra known as recursive substitution– Consider the value of y at time t = 1, then its
value at time t = 2 and so on– These values are:
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
1 0 1
22 1 2 0 1 2 0 1 2
21 2 0
( )
..... tt t t t
y y v
y y v y v v y v v
y v v v y
Principles of Econometrics, 4th Edition
Page 17Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The mean of yt is:
Real-world data rarely have a zero mean –We can introduce a nonzero mean μ as:
– Or
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
21 2 0t t t tE y E v v v
1( ) ( )t t ty y v
1 , 1t t ty y v Eq. 12.2b
Principles of Econometrics, 4th Edition
Page 18Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
With α = 1 and ρ = 0.7:
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
( ) / (1 ) 1 / (1 0.7) 3.33tE y
Principles of Econometrics, 4th Edition
Page 19Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
An extension to Eq. 12.2a is to consider an AR(1) model fluctuating around a linear trend: (μ + δt)
– Let the ‘‘de-trended’’ series (yt -μ - δt) behave like an autoregressive model:
Or:
12.1Stationary and Nonstationary
Variables
12.1.1The First-Order Autoregressive
Model
1( ) ( ( 1)) , 1 t t ty t y t v
1t t ty y t v Eq. 12.2c
Principles of Econometrics, 4th Edition
Page 20Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider the special case of ρ = 1:
– This model is known as the random walk model• These time series are called random walks
because they appear to wander slowly upward or downward with no real pattern• the values of sample means calculated from
subsamples of observations will be dependent on the sample period–This is a characteristic of nonstationary
series
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
Eq. 12.3a 1t t ty y v
Principles of Econometrics, 4th Edition
Page 21Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
We can understand the ‘‘wandering’’ by recursive substitution:
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1 0 1
2
2 1 2 0 1 2 01
1 01
( )
ss
t
t t t ss
y y v
y y v y v v y v
y y v y v
Principles of Econometrics, 4th Edition
Page 22Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The term is often called the stochastic trend– This term arises because a stochastic
component vt is added for each time t, and because it causes the time series to trend in unpredictable directions
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1
t
ssv
Principles of Econometrics, 4th Edition
Page 23Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Recognizing that the vt are independent, taking the expectation and the variance of yt yields, for a fixed initial value y0:
– The random walk has a mean equal to its initial value and a variance that increases over time, eventually becoming infinite
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
0 1 2 0( ) ( ... )t tE y y E v v v y
21 2var( ) var( ... ) σt t vy v v v t
Principles of Econometrics, 4th Edition
Page 24Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Another nonstationary model is obtained by adding a constant term:
– This model is known as the random walk with drift
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1t t ty y v Eq. 12.3b
Principles of Econometrics, 4th Edition
Page 25Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
A better understanding is obtained by applying recursive substitution:
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1 0 1
2
2 1 2 0 1 2 01
1 01
( ) 2
ss
t
t t t ss
y y v
y y v y v v y v
y y v t y v
Principles of Econometrics, 4th Edition
Page 26Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The term tα a deterministic trend component– It is called a deterministic trend because a fixed
value α is added for each time t – The variable y wanders up and down as well as
increases by a fixed amount at each time t
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
Principles of Econometrics, 4th Edition
Page 27Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The mean and variance of yt are:
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
0 1 2 3 0( ) ( ... )t tE y t y E v v v v t y 2
1 2 3var( ) var( ... )t t vy v v v v t
Principles of Econometrics, 4th Edition
Page 28Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
We can extend the random walk model even further by adding a time trend:
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1t t ty t y v Eq. 12.3c
Principles of Econometrics, 4th Edition
Page 29Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The addition of a time-trend variable t strengthens the trend behavior:
where we used:
12.1Stationary and Nonstationary
Variables
12.1.2Random Walk
Models
1 0 1
2
2 1 2 0 1 2 01
1 01
2 2 ( ) 2 3
( 1)
2
ss
t
t t t ss
y y v
y y v y v v y v
t ty t y v t y v
1 2 3 1 2t t t
Principles of Econometrics, 4th Edition
Page 30Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.2
Spurious Regressions
Principles of Econometrics, 4th Edition
Page 31Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The main reason why it is important to know whether a time series is stationary or nonstationary before one embarks on a regression analysis is that there is a danger of obtaining apparently significant regression results from unrelated data when nonstationary series are used in regression analysis – Such regressions are said to be spurious
12.2Spurious
Regressions
Principles of Econometrics, 4th Edition
Page 32Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider two independent random walks:
– These series were generated independently and, in truth, have no relation to one another
– Yet when plotted we see a positive relationship between them
12.2Spurious
Regressions
1 1 1
2 1 2
: :
t t t
t t t
rw y y vrw x x v
Principles of Econometrics, 4th Edition
Page 33Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.2Spurious
Regressions FIGURE 12.3 Time series and scatter plot of two random walk variables
Principles of Econometrics, 4th Edition
Page 34Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.2Spurious
Regressions
FIGURE 12.3 (Continued) Time series and scatter plot of two random walk variables
Principles of Econometrics, 4th Edition
Page 35Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
A simple regression of series one (rw1) on series two (rw2) yields:
– These results are completely meaningless, or spurious • The apparent significance of the relationship
is false
12.2Spurious
Regressions
21 217.818 0.842 , 0.70
( ) (40.837)t trw rw R
t
Principles of Econometrics, 4th Edition
Page 36Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
When nonstationary time series are used in a regression model, the results may spuriously indicate a significant relationship when there is none– In these cases the least squares estimator and
least squares predictor do not have their usual properties, and t-statistics are not reliable
– Since many macroeconomic time series are nonstationary, it is particularly important to take care when estimating regressions with macroeconomic variables
12.2Spurious
Regressions
Principles of Econometrics, 4th Edition
Page 37Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.3
Unit Root Tests for Stationarity
Principles of Econometrics, 4th Edition
Page 38Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
There are many tests for determining whether a series is stationary or nonstationary– The most popular is the Dickey–Fuller test
12.3Unit Root Tests for
Stationarity
Principles of Econometrics, 4th Edition
Page 39Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The AR(1) process yt = ρyt-1 + vt is stationary when |ρ| < 1– But, when ρ = 1, it becomes the nonstationary
random walk process–We want to test whether ρ is equal to one or
significantly less than one• Tests for this purpose are known as unit root
tests for stationarity
12.3Unit Root Tests for
Stationarity
12.3.1Dickey-Fuller Test 1 (No constant and No
Trend)
Principles of Econometrics, 4th Edition
Page 40Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider again the AR(1) model:
–We can test for nonstationarity by testing the null hypothesis that ρ = 1 against the alternative that |ρ| < 1• Or simply ρ < 1
12.3Unit Root Tests for
Stationarity
1t t ty y v Eq. 12.4
12.3.1Dickey-Fuller Test 1 (No constant and No
Trend)
Principles of Econometrics, 4th Edition
Page 41Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
A more convenient form is:
– The hypotheses are:
12.3Unit Root Tests for
Stationarity
Eq. 12.5a 1 1 1
1
1
1t t t t t
t t t
t t
y y y y v
y y v
y v
0 0
1 1
: 1 : 0
: 1 : 0
H H
H H
12.3.1Dickey-Fuller Test 1 (No constant and No
Trend)
Principles of Econometrics, 4th Edition
Page 42Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The second Dickey–Fuller test includes a constant term in the test equation:
– The null and alternative hypotheses are the same as before
12.3Unit Root Tests for
Stationarity
12.3.2Dickey-Fuller Test 2 (With Constant but
No Trend)
Eq. 12.5b 1t t ty y v
Principles of Econometrics, 4th Edition
Page 43Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The third Dickey–Fuller test includes a constant and a trend in the test equation:
– The null and alternative hypotheses are H0: γ = 0 and H1:γ < 0 (same as before)
12.3Unit Root Tests for
Stationarity
12.3.3Dickey-Fuller Test 3 (With Constant and
With Trend)
Eq. 12.5c 1t t ty y t v
Principles of Econometrics, 4th Edition
Page 44Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
To test the hypothesis in all three cases, we simply estimate the test equation by least squares and examine the t-statistic for the hypothesis that γ = 0 – Unfortunately this t-statistic no longer has the
t-distribution– Instead, we use the statistic often called a τ
(tau) statistic
12.3Unit Root Tests for
Stationarity
12.3.4The Dickey-Fuller
Critical Values
Principles of Econometrics, 4th Edition
Page 45Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.3Unit Root Tests for
Stationarity
12.3.4The Dickey-Fuller
Critical Values
Table 12.2 Critical Values for the Dickey–Fuller Test
Principles of Econometrics, 4th Edition
Page 46Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
To carry out a one-tail test of significance, if τc is the critical value obtained from Table 12.2, we reject the null hypothesis of nonstationarity if τ ≤ τc
– If τ > τc then we do not reject the null hypothesis that the series is nonstationary
12.3Unit Root Tests for
Stationarity
12.3.4The Dickey-Fuller
Critical Values
Principles of Econometrics, 4th Edition
Page 47Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
An important extension of the Dickey–Fuller test allows for the possibility that the error term is autocorrelated– Consider the model:
where
12.3Unit Root Tests for
Stationarity
12.3.4The Dickey-Fuller
Critical Values
11
m
t t s t s ts
y y a y v
Eq. 12.6
1 1 2 2 2 3, ,t t t t t ty y y y y y
Principles of Econometrics, 4th Edition
Page 48Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The unit root tests based on Eq. 12.6 and its variants (intercept excluded or trend included) are referred to as augmented Dickey–Fuller tests–When γ = 0, in addition to saying that the series
is nonstationary, we also say the series has a unit root
– In practice, we always use the augmented Dickey–Fuller test
12.3Unit Root Tests for
Stationarity
12.3.4The Dickey-Fuller
Critical Values
Principles of Econometrics, 4th Edition
Page 49Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.3Unit Root Tests for
Stationarity
12.3.5The Dickey-Fuller Testing Procedures
Table 12.3 AR processes and the Dickey-Fuller Tests
Principles of Econometrics, 4th Edition
Page 50Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The Dickey-Fuller testing procedure:– First plot the time series of the variable and select a
suitable Dickey-Fuller test based on a visual inspection of the plot• If the series appears to be wandering or
fluctuating around a sample average of zero, use test equation (12.5a)• If the series appears to be wandering or
fluctuating around a sample average which is nonzero, use test equation (12.5b)• If the series appears to be wandering or
fluctuating around a linear trend, use test equation (12.5c)
12.3Unit Root Tests for
Stationarity
12.3.5The Dickey-Fuller Testing Procedures
Principles of Econometrics, 4th Edition
Page 51Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The Dickey-Fuller testing procedure (Continued):– Second, proceed with one of the unit root tests
described in Sections 12.3.1 to 12.3.3
12.3Unit Root Tests for
Stationarity
12.3.5The Dickey-Fuller Testing Procedures
Principles of Econometrics, 4th Edition
Page 52Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
As an example, consider the two interest rate series:
– The federal funds rate (Ft)
– The three-year bond rate (Bt)
Following procedures described in Sections 12.3 and 12.4, we find that the inclusion of one lagged difference term is sufficient to eliminate autocorrelation in the residuals in both cases
12.3Unit Root Tests for
Stationarity
12.3.6The Dickey-Fuller Tests: An Example
Principles of Econometrics, 4th Edition
Page 53Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The results from estimating the resulting equations are:
– The 5% critical value for tau (τc) is -2.86
– Since -2.505 > -2.86, we do not reject the null hypothesis of non-stationarity.
12.3Unit Root Tests for
Stationarity
12.3.6The Dickey-Fuller Tests: An Example
1 1
1 1
0.173 0.045 0.561
( ) ( 2.505)
0.237 0.056 0.237
( ) ( 2.703)
t t t
t t t
F F F
tau
B B B
tau
Principles of Econometrics, 4th Edition
Page 54Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Recall that if yt follows a random walk, then γ = 0 and the first difference of yt becomes:
– Series like yt, which can be made stationary by taking the first difference, are said to be integrated of order one, and denoted as I(1)• Stationary series are said to be integrated of
order zero, I(0)– In general, the order of integration of a series is
the minimum number of times it must be differenced to make it stationary
12.3Unit Root Tests for
Stationarity
12.3.7Order of Integration
1t t t ty y y v
Principles of Econometrics, 4th Edition
Page 55Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The results of the Dickey–Fuller test for a random walk applied to the first differences are:
12.3Unit Root Tests for
Stationarity
12.3.7Order of Integration
10.447
( ) ( 5.487)
t tF F
tau
10.701
( ) ( 7.662)
t tB B
tau
Principles of Econometrics, 4th Edition
Page 56Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Based on the large negative value of the tau statistic (-5.487 < -1.94), we reject the null hypothesis that ΔFt is nonstationary and accept the alternative that it is stationary
–We similarly conclude that ΔBt is stationary (-7.662 < -1.94)
12.3Unit Root Tests for
Stationarity
12.3.7Order of Integration
Principles of Econometrics, 4th Edition
Page 57Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.4
Cointegration
Principles of Econometrics, 4th Edition
Page 58Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
As a general rule, nonstationary time-series variables should not be used in regression models to avoid the problem of spurious regression– There is an exception to this rule
12.4Cointegration
Principles of Econometrics, 4th Edition
Page 59Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
There is an important case when et = yt - β1 - β2xt is a stationary I(0) process
– In this case yt and xt are said to be cointegrated
• Cointegration implies that yt and xt share similar stochastic trends, and, since the difference et is stationary, they never diverge too far from each other
12.4Cointegration
Principles of Econometrics, 4th Edition
Page 60Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The test for stationarity of the residuals is based on the test equation:
– The regression has no constant term because the mean of the regression residuals is zero.
–We are basing this test upon estimated values of the residuals
12.4Cointegration
1ˆ ˆγt t te e v Eq. 12.7
Principles of Econometrics, 4th Edition
Page 61Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.4Cointegration Table 12.4 Critical Values for the Cointegration Test
Principles of Econometrics, 4th Edition
Page 62Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
There are three sets of critical values–Which set we use depends on whether the
residuals are derived from:
12.4Cointegration
Eq. 12.8a ˆ1: t t tEquation e y bx
2 1ˆ2 : t t tEquation e y b x b
2 1ˆˆ3: t t tEquation e y b x b t
Eq. 12.8b
Eq. 12.8c
Principles of Econometrics, 4th Edition
Page 63Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider the estimated model:
– The unit root test for stationarity in the estimated residuals is:
12.4Cointegration
Eq. 12.9
12.4.1An Example of a
Cointegration Test
2ˆ 1.140 0.914 , 0.881
( ) (6.548) (29.421)t tB F R
t
1 1ˆ ˆ ˆ0.225 0.254
( ) ( 4.196)t t te e e
tau
Principles of Econometrics, 4th Edition
Page 64Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The null and alternative hypotheses in the test for cointegration are:
– Similar to the one-tail unit root tests, we reject the null hypothesis of no cointegration if τ ≤ τc, and we do not reject the null hypothesis that the series are not cointegrated if τ > τc
12.4Cointegration
12.4.1An Example of a
Cointegration Test
0
1
: the series are not cointegrated residuals are nonstationary
: the series are cointegrated residuals are stationary
H
H
Principles of Econometrics, 4th Edition
Page 65Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider a general model that contains lags of y and x– Namely, the autoregressive distributed lag
(ARDL) model, except the variables are nonstationary:
12.4Cointegration
12.4.2The Error Correction
Model
1 1 0 1 1δ θ δ δt t t t ty y x x v
Principles of Econometrics, 4th Edition
Page 66Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
If y and x are cointegrated, it means that there is a long-run relationship between them– To derive this exact relationship, we set
yt = yt-1 = y, xt = xt-1 = x and vt = 0
– Imposing this concept in the ARDL, we obtain:
• This can be rewritten in the form:
12.4Cointegration
12.4.2The Error Correction
Model
1 0 11 θ δ δ δy x
1 2β βy x
Principles of Econometrics, 4th Edition
Page 67Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Add the term -yt-1 to both sides of the equation:
– Add the term – δ0xt-1+ δ0xt-1:
–Manipulating this we get:
12.4Cointegration
12.4.2The Error Correction
Model
1 1 1 0 1 1δ θ 1 δ δt t t t t ty y y x x v
1 1 0 1 0 1 1δ θ 1 δ δ δt t t t t ty y x x x v
0 11 1 1 0
1 1
δ δδθ 1 δ
θ 1 θ 1t t t t ty y x x v
Principles of Econometrics, 4th Edition
Page 68Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Or:
– This is called an error correction equation– This is a very popular model because:• It allows for an underlying or fundamental
link between variables (the long-run relationship)• It allows for short-run adjustments (i.e.
changes) between variables, including adjustments to achieve the cointegrating relationship
12.4Cointegration
12.4.2The Error Correction
Model
1 1 2 1 0α β β δt t t t ty y x x v Eq. 12.10
Principles of Econometrics, 4th Edition
Page 69Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
For the bond and federal funds rates example, we have:
– The estimated residuals are
12.4Cointegration
12.4.2The Error Correction
Model
1 1 1ˆ 0.142 1.429 0.777 0.842 0.327
2.857 9.387 3.855
t t t t tB B F F F
t
1 1 1ˆ 1.429 0.777t t te B F
Principles of Econometrics, 4th Edition
Page 70Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
The result from applying the ADF test for stationarity is:
– Comparing the calculated value (-3.929) with the critical value, we reject the null hypothesis and conclude that (B, F) are cointegrated
1 1ˆ ˆ ˆ0.169 0.180
3.929t t te e e
t
12.4Cointegration
Principles of Econometrics, 4th Edition
Page 71Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.5
Regression with No-Cointegration
Principles of Econometrics, 4th Edition
Page 72Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
How we convert nonstationary series to stationary series, and the kind of model we estimate, depend on whether the variables are difference stationary or trend stationary– In the former case, we convert the
nonstationary series to its stationary counterpart by taking first differences
– In the latter case, we convert the nonstationary series to its stationary counterpart by de-trending
12.5Regression When
There is No Cointegration
Principles of Econometrics, 4th Edition
Page 73Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider the random walk model:
– This can be rendered stationary by taking the first difference:
• The variable yt is said to be a first difference stationary series
12.5Regression When
There is No Cointegration
12.5.1First Difference
Stationary
1t t ty y v
1t t t ty y y v
Principles of Econometrics, 4th Edition
Page 74Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
A suitable regression involving only stationary variables is:
– Now consider a series yt that behaves like a random walk with drift:
with first difference:
• The variable yt is also said to be a first difference stationary series, even though it is stationary around a constant term
12.5Regression When
There is No Cointegration
12.5.1First Difference
Stationary
1 0 1 1t t t t ty y x x e Eq. 12.11a
1t t ty y v
t ty v
Principles of Econometrics, 4th Edition
Page 75Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Suppose that y and x are I(1) and not cointegrated– An example of a suitable regression equation is:
12.5Regression When
There is No Cointegration
12.5.1First Difference
Stationary
Eq. 12.11b 1 0 1 1t t t t ty y x x e
Principles of Econometrics, 4th Edition
Page 76Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Consider a model with a constant term, a trend term, and a stationary error term:
– The variable yt is said to be trend stationary because it can be made stationary by removing the effect of the deterministic (constant and trend) components:
12.5Regression When
There is No Cointegration
12.5.2Trend Stationary
t ty t v
t ty t v
Principles of Econometrics, 4th Edition
Page 77Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
If y and x are two trend-stationary variables, a possible autoregressive distributed lag model is:
12.5Regression When
There is No Cointegration
12.5.2Trend Stationary
Eq. 12.12 1 0 1 1t t t t ty y x x e
Principles of Econometrics, 4th Edition
Page 78Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
As an alternative to using the de-trended data for estimation, a constant term and a trend term can be included directly in the equation:
where:
12.5Regression When
There is No Cointegration
12.5.2Trend Stationary
1 0 1 1t t tt ty t y x x e
1 1 2 0 1 1 1 1 2(1 ) ( )
1 1 2 0 1(1 ) ( )
Principles of Econometrics, 4th Edition
Page 79Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression
If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term
If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable
12.5Regression When
There is No Cointegration
12.5.3Summary
Principles of Econometrics, 4th Edition
Page 80Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
12.5Regression When
There is No Cointegration
12.5.3Summary
FIGURE 12.4 Regression with time-series data: nonstationary variables
Principles of Econometrics, 4th Edition
Page 81Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
Key Words
Principles of Econometrics, 4th Edition
Page 82Chapter 12: Regression with Time-Series Data:
Nonstationary Variables
autoregressive process
cointegration
Dickey–Fuller tests
difference stationary
mean reversion
nonstationary
Keywords
order of integration
random walk process
random walk with drift
spurious regressions
stationary
stochastic process
stochastic trend
tau statistic
trend stationary
unit root tests