Lecture 18. Serial correlation: testing and estimation Testing for serial correlation In lecture 16 we used graphical methods to look for serial/autocorrelation in the random error term t u . Because we cannot observe the t u we used the OLS residuals t e . We looked at • Time series graph of n t e t , , 1 , … = . If there is serial correlation this graph shows gradual changes in the t e . • Scatterplot of t e versus 1 − t e . If the AR(1) model t t t u u ε ρ + = − 1 holds, then we expect that the scatterplot is concentrated along a straight line through 0. Tests for serial/autocorrelation also use the OLS residuals t e .
Transcript
Lecture 18. Serial correlation: testing and estimation Testing for serial correlation In lecture 16 we used graphical methods to look for serial/autocorrelation in the random error term tu . Because we cannot observe the tu we used the OLS residuals te . We looked at
• Time series graph of ntet ,,1, …= . If there is serial correlation this graph shows gradual changes in the te .
• Scatterplot of te versus 1−te . If the AR(1) model ttt uu ερ += −1 holds, then we expect that the scatterplot is concentrated along a straight line through 0.
Tests for serial/autocorrelation also use the OLS residuals te .
Consider the linear regression model
ntuXXY ttKKtt ,,1,221 KL =+++= βββ As with tests for heteroskedasticity we assume a particular model for the autocorrelation. Initially we consider first-order serial correlation (1) ttt uu ερ += −1 11 <<− ρ with tε white noise (the tε ’s are independent and have all the same variance and mean 0). If 0=ρ , then ttu ε= and in that case the random errors tu satisfy Assumption 4, i.e. there is no serial correlation. Hence a test for serial correlation is a test of
0:0 =ρH .
First step is to find estimator for ρ . If we replace tu in (1) by te and estimate ρ by OLS we obtain
∑
∑
=
=−
=n
tt
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ttt
e
ee
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ρ̂
This is also the first-order autocorrelation coefficient of the time series ntet ,,1, K= (see lecture 16) . The obvious thing to do is to use ρ̂ to test whether 0=ρ . Instead of ρ̂ , a related quantity is used, the Durbin-Watson statistic d
∑
∑
=
=−−
=n
tt
n
ttt
e
ee
d
1
2
2
21)(
It can be shown )ˆ1(2 ρ−=d Hence if ρ̂ close to 0 (no autocorrelation) then d is close to 2. If ρ̂ is close to 1, then d is close to 0 and if ρ̂ is close to –1, then d is close to 4 positive no negative autocor. autocor. autocor. 0 2 4 ___|____|__|__|_____|______________|__ Lc c Uc d Because negative autocorrelation is rare, the usual test is 0:0 =ρH against 0:1 >ρH . We reject (see graph) if d is small, i.e. close to 0. The critical value is than some number c greater than 0 but less than 2.
With a 5% significance level we want to have that if 0H is true, then the probabability of rejection of 0H
05.)Pr( =< cd If the errors have a normal distribution (assumption 5), then the distribution of d can be derived. This distribution depends on the independent variables KXX ,,2 K . Compare with the t- or F-distribution that do not depend on this. Some programs compute c exactly for the independent variables in your dataset. This is easy with current computers. If not, then there is a table with bounds Lc and Uc . These bounds are for ‘extreme’ datasets and the c for any dataset is between them.
Example: for 2 independent variables (do not count the constant) and 25 observations
206.1=Lc and 550.1=Uc . Hence if e.g. 1.1=d we reject and if 7.1=d we do not reject. If
3.1=d we do know what to do (test is inconclusive). This is computational problem, because c can be computed. Consider regression log housing starts per head on log GNP per head and log mortgage interest rate The DW statistics is .913 with n=23, k’=2, so that 168.1=Lc and we reject the hypothesis of no serial correlation.
R-squared 0.094147 Mean dependent var 1.991961Adjusted R-squared 0.003562 S.D. dependent var 0.226095S.E. of regression 0.225692 Akaike info criterion -0.018186Sum squared resid 1.018735 Schwarz criterion 0.129922Log likelihood 3.209133 F-statistic 1.039325Durbin-Watson stat 0.913015 Prob(F-statistic) 0.372027
Alternative to DW test is the Lagrange Multiplier (LM) test. Also uses the OLS residuals te . The first step of the test is a linear regression with dependent variable te and independent variables 12 ,,, −ttKt eXX K . Compute the 2R of this regression. The test statistic is 2)1( RnLM −= Note that we use 1−n observations in the regression. If 0:0 =ρH is true than LM has a chi-square distribution with 1 d.f. We reject if
cLM > and if we want a test with a 5% significance level we find the critical value c from 05.)Pr( => cLM
Application to housing start data 6.85311.*22 ==LM and the critical value for 5% significance is 3.84. Again we reject 0H . Estimation with serial correlation Consider the linear regression
ttt uXY ++= 21 ββ and ttt uu ερ += −1 AR(1) How do we estimate the regression parameters and ρ ? As with heteroskedasticity we transform the variables such that we have a random error term that satisfies the assumptions 1-4. Hence we can apply OLS to the transformed regression.
Dependent Variable: RESID01Method: Least SquaresDate: 11/14/01 Time: 23:56Sample(adjusted): 1964 1985Included observations: 22 after adjusting endpoints
R-squared 0.311452 Mean dependent var -0.006314Adjusted R-squared 0.196694 S.D. dependent var 0.218061S.E. of regression 0.195443 Akaike info criterion -0.264134Sum squared resid 0.687561 Schwarz criterion -0.065763Log likelihood 6.905473 F-statistic 2.713985Durbin-Watson stat 1.272797 Prob(F-statistic) 0.075366
Because tε satisfies all the usual assumptions we must get this as the random error term. Note 1−−= ttt uu ρε Now do the subtraction ttt uXY ++= 21 ββ 11211 −−− ++= ttt uXY ρρβρβρ _______________________ (1) ttttt XXYY ερββρρ +−+−=− −− )()1( 1211 Conclusion: if we transform the dependent variable to 1−− tt YY ρ and the independent variable to 1−− tt XX ρ we can use OLS to estimate 2β . Note that the OLS estimator of the constant does not estimate 1β , but if we divide the OLS estimator of the constant by ρ−1 we get an estimator of 1β .
Problem with this method: We do not know ρ . Solution: Choose range of values for ρ , e.g. -.99, -.98, …., .98, .99 and estimate (1) for each of these values. For each ρ compute the residuals )()1( 1211 −− −−−−−= ttttt XXYYe ρββρρ and the sum of squared residuals. Choose the value of ρ and the OLS estimators of 21, ββ that has the smallest sum of squared residuals. This the Hildreth-Lu procedure. Application to consumption and wages (billion 1992$) for US 1959-1994.
• Test for AR(1) errors (DW and LM) • Compare estimates and standard errors
C 1614.711 59.81012 26.99729 0.0000WAGES 0.769682 0.030647 25.11440 0.0000
R-squared 0.948852 Mean dependent var 2811.178Adjusted R-squared 0.947347 S.D. dependent var 945.5435S.E. of regression 216.9661 Akaike info criterion 13.65131Sum squared resid 1600526. Schwarz criterion 13.73929Log likelihood -243.7236 F-statistic 630.7330Durbin-Watson stat 0.072411 Prob(F-statistic) 0.000000
Dependent Variable: RESID01Method: Least SquaresDate: 11/15/01 Time: 01:12Sample(adjusted): 1960 1994Included observations: 35 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 48.97566 13.26937 3.690880 0.0008WAGES -0.026748 0.006724 -3.978251 0.0004
RESID01LAG 0.930118 0.037631 24.71658 0.0000
R-squared 0.950467 Mean dependent var 12.50131Adjusted R-squared 0.947371 S.D. dependent var 203.1813S.E. of regression 46.61194 Akaike info criterion 10.60341Sum squared resid 69525.55 Schwarz criterion 10.73672Log likelihood -182.5596 F-statistic 307.0144Durbin-Watson stat 1.096482 Prob(F-statistic) 0.000000
Dependent Variable: CONSMethod: Least SquaresDate: 11/15/01 Time: 01:13Sample(adjusted): 1960 1994Included observations: 35 after adjusting endpointsConvergence achieved after 8 iterations
Variable Coefficient Std. Error t-Statistic Prob.
C 2566.421 932.1594 2.753200 0.0096WAGES 0.519497 0.138746 3.744220 0.0007
AR(1) 0.943172 0.047851 19.71054 0.0000
R-squared 0.997574 Mean dependent var 2851.680Adjusted R-squared 0.997423 S.D. dependent var 927.1223S.E. of regression 47.06673 Akaike info criterion 10.62283Sum squared resid 70888.86 Schwarz criterion 10.75614Log likelihood -182.8995 F-statistic 6580.217Durbin-Watson stat 1.135516 Prob(F-statistic) 0.000000
Inverted AR Roots .94
Alternative interpretation of AR(1) errors The linear regression in (1) can be rewritten as (2) ttttt XXYY ερββρβρ +−++−= −− 12211)1(
This is a linear regression model with independent variables 11 ,, −− ttt XXY . In the model with only tX as independent variable 11, −− tt XY are omitted and relegated to the error term. Because both variables are economic time series and change gradually the error term is autocorrelated. Compare (2) to the linear regression model (3) ttttt XXYY εγγγγ ++++= −− 143121 Note that (3) has 4 regression coefficients and
(2) has 3. (3) becomes (2) if 23
4 γγγ
−= .
If we estimate (3) we find 691.ˆ 4 −=γ , 718.ˆ3 =γ , 933.ˆ2 =γ and hence
962.718.
691.
ˆ
ˆ
3
4 −=−
=γγ
Note that (3) is more general and an alternative to (2).
Dependent Variable: CONSMethod: Least SquaresDate: 11/15/01 Time: 01:15Sample(adjusted): 1960 1994Included observations: 35 after adjusting endpoints
