Problem Set 8 Answers: Autocorrelation

Transcription

1 Problem Set 8 Answers: Autocorrelation. u uv /* read in data set contains 24 annual observations on vacancy numbers and the unemployment rate */. tsset year /* set up time variable. regdw v u time variable: year, 75 to 98 Source SS df MS Number of obs = F( 1, 22) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = vacant Coef. Std. Err. t P> t [95% Conf. Interval] unemp _cons Durbin-Watson Statistic = From Tables given sample size T =24 and k = 1 (number of rhs parameters excluding constant) dlow = and dhigh = so estimated Durbin-Watson is > dhigh so accept null of no autocorrelation in residuals Check by graphing residuals over time. predict resuv, resid /* save residuals from previous regression */. gra resuv year, yline(0) ylab xlab(75,85,95,98) /* graph them */ 4 2 Residuals year Little evidence of systematic variation

2 Now Durbin-Watson test is potentially ambiguous and is not valid in the presence of endogenous variables (eg lagged dependent variables). In this case might be tempted to use Breusch-Godfrey test. The latter test can also be used for tests of autocorrelation of any lag length.. predict res, resid. g res1=res[_n-1]. g res2=res[_n-2]. g res3=res[_n-3]. g res4=res[_n-4] Test for 1 st order autocorrelation is to save residuals from original specification, lag them 1 period and then regress residuals on the lag and the rhs variable(s) in the original regression (in this case u). No autocorrelation should mean the lagged residuals have no explanatory power in this auxiliary regression (the x variables are there to net out any influence on the lagged residuals that could otherwise show up as significant on the residual lags).. reg res res1 u Source SS df MS Number of obs = F( 2, 20) = 0.03 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = res Coef. Std. Err. t P> t [95% Conf. Interval] res unemp _cons and (N-q)*R 2 aux = (24-1)*.003 = (Note N-q = number of observations in auxiliary regression) This statistic has a chi-squared distribution with 1 degree of freedom (equal to the number of lags tested) From tables χ 2 critical at 5% level = 3.84 So estimated χ 2 < χ 2 critical, so as before, accept null that residuals are not correlated over one year to the next. For test of AR(4). reg res res1 res2 res3 res4 u Source SS df MS Number of obs = F( 5, 14) = 0.96 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = res Coef. Std. Err. t P> t [95% Conf. Interval]

3 res res res res unemp _cons and (N-q)*R 2 aux = 20*.255 = 5.1 This statistic has a chi-squared distribution with 4 degree of freedom (equal to the number of lags tested) From tables χ 2 critical at 5% level = So estimated χ 2 < χ 2 critical, so accept null that residuals are not correlated over a 4 year period year to the next. Remember, however that Breusch-Godfrey test is only valid asymptotically, and sample size is far from large here, so would prefer results from Durbin-Watson Sometimes, taking logs can reduce autocorrelation. In the example below, however, taking logs makes things worse.. g logv=log(vacant) /* log of vacancies */. g logu=log(unemp) /* log of unemployment */. regdw logv logu Source SS df MS Number of obs = F( 1, 22) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = logv Coef. Std. Err. t P> t [95% Conf. Interval] logu _cons Durbin-Watson Statistic = So given same sample size and number of parameters as before, critical values for Durbin-Watson test must be the same ie dlow = and dhigh = so estimated Durbin-Watson is now < dlow so reject null of no autocorrelation in residuals and conclude there is positive autocorrelation in residuals. Remember different functional forms can generate different degrees of autocorrelation in residuals, so always wise to experiment with different functional forms when choosing models. Using fact the DW 2(1-?) Can estimate? as (2-DW)/2 = (2-1.09)/2 =.455 Which says around half the level of last year s residual is carried over into the estimated level of this year s residual, (hence highly positively autocorrelated) To remove autocorrelation could try writing down the FGLS specification.

4 2. Now read in investment data containing 30 annual observations. u invest. tsset year time variable: year, 60 to 89. regdw invest GNP interest Source SS df MS Number of obs = F( 2, 27) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = invest Coef. Std. Err. t P> t [95% Conf. Interval] GNP interest _cons Durbin-Watson Statistic = So given same sample size (30) and number of parameters less constant k = 3-1 = 2, so critical values for Durbin-Watson test are now (see Tables) dlow = and dhigh = so estimated Durbin-Watson is < dlow so reject null of no autocorrelation in residuals and conclude there is positive autocorrelation in residuals.. predict res, resid. gra res year, yline(0) ylab xlab(60,70,80,90) 5 0 Residuals year Suggests periods of positive residuals (1960 s), followed by negative residuals (1970s) followed by positive (1980s), ie symptom of positive autocorrelation.

5 To do Breusch-Godfrey test automatically in Stata, type the following commands. reg invest GNP interest Source SS df MS Number of obs = F( 2, 27) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = invest Coef. Std. Err. t P> t [95% Conf. Interval] GNP interest _cons bgtest, lags(1) Breusch-Godfrey LM statistic: Chi-sq( 1) P-value = bgtest, lags(4) Breusch-Godfrey LM statistic: Chi-sq( 4) P-value = i) See lecture notes ii) From information in Table, sample size = 50 ( is 50 observations) So DW critical values are for T = 50 and k = 1 (k value in test is number of (From Tables) d low = 1.50 d upper = 1.59 so in 1 st regression DW < d low which indicates presence of (positive) 1 st order autocorrelation. Hence OLS coefficients in column 1 will be unbiased but standard errors and therefore t and F values and confidence intervals are all invalid. (Consequence: may mistakenly conclude variable is statistically significant from zero, for example, when it is not). 2 nd regression may be an improvement (including lags of the explanatory variables is sometimes one way of reducing autocorrelation) Now DW critical values are for T = 50 and k = 2 (k value in test is number of (From Tables) d low = 1.46 d upper = 1.63 Since DW lies between critical values, test is inconclusive 4. i) Use Durbin-Watson Test, sample size = 40, k =1 (k value in test is number of So DW critical values are for T = 40 and k = 1 (k value in test is number of (From Tables) d low = 1.44 d upper = 1.54

6 so in 1 st regression DW < d low which indicates presence of (positive) 1 st order autocorrelation. Hence OLS coefficients in column 1 will be unbiased but standard errors and hence t and F statistics will be biased. Breusch-Godfrey test gives (N-q)*R 2 aux = 40-1*.2 = 7.8 Since this is a test of one lag then this statistic has a chi-squared distribution with 1 degree of freedom (equal to the number of lags tested) From tables χ 2 critical at 5% level = 3.84 So estimated χ 2 > χ 2 critical, so as before, reject null that residuals are not correlated over one year to the next. 5. i) Use Durbin-Watson Test, sample size = 60, k =1 (k value in test is number of So DW critical values are for T = 60 and k = 1 (k value in test is number of (From Tables) d low = 1.55 d upper = 1.62 so in 1 st regression DW < d low which indicates presence of (positive) 1 st order autocorrelation. Hence OLS coefficients in column 1 will be unbiased but standard errors and hence t and F statistics will be biased. Again use Durbin-Watson Test, sample size = 60, k =2 (k value in test is number of So DW critical values are for T = 60 and k = 2 (k value in test is number of (From Tables) d low = 1.51 d upper = 1.65 so in 1 st regression DW > d upper which suggests no (positive) 1 st order autocorrelation. But We know DW is biased toward 2 in presence of lagged dependent variables, so can t rely on this. Try instead, (since sample size relatively small also can t rely on Breusch-Godfrey) Durbin s h test From Table h>1.96 which means reject null of no autocorrelation