Tutorial 2: Autocorrelation Matthew Robson University of York Econometrics 2

Autocorrelation Autocorrelation emerges when the errors in different time-periods are correlated. When 𝑐𝑜𝑣 𝑢 𝑖 , 𝑢 𝑗 ≠0, 𝑖≠𝑗 Positive Autocorrelation Negative Autocorrelation (Gujarati and Porter, 2009)

Assignment 7 Estimate the log-linear consumption function: 𝐿𝑜𝑔 𝐶 𝑡 = 𝛽 0 + 𝛽 1 𝐿𝑜𝑔 𝐼 𝑡 + 𝛽 2 𝐿𝑜𝑔 𝑊 𝑡 + 𝛽 4 𝑟 𝑡 + 𝑢 𝑡 Where: 𝐶 𝑡 = consumption, 𝐼 𝑡 = real disposable income 𝑊 𝑡 = wealth, 𝑟 𝑡 = interest rate For the period 1967q1 – 2002q4. (1)

Descriptive Statistics

Results 𝐿𝑜𝑔 𝐶 𝑡 =0.8559+0.9165 𝐿𝑜𝑔 𝐼 𝑡 +0.0138 𝐿𝑜𝑔 𝑊 𝑡 +−0.0022 𝑟 𝑡 + 𝑢 𝑡

Predicted Values

Autocorrelation

Question a) Test for autocorrelation using the Durbin-Watson test statistic given by PC-GIVE. What are the limitations of this test? How does the Breusch-Godfrey test overcome some of these limitations?

Durbin-Watson Test Defined as: 𝑑= 𝑡=2 𝑡=𝑛 𝑢 𝑡 − 𝑢 𝑡−1 2 𝑡=1 𝑡=𝑛 𝑢 𝑡 2

Durbin-Watson Test Durbin-Watson Statistic: 𝑑 = 0.439 𝑁=144, 𝑘 = 3 𝑀𝑜𝑑𝑒𝑙→ 𝑇𝑒𝑠𝑡→ 𝑇𝑒𝑠𝑡…→𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛𝑠, 𝑃𝑜𝑟𝑡𝑚𝑎𝑛𝑡𝑒𝑎𝑢 𝑎𝑛𝑑 𝐷𝑊 Durbin-Watson Statistic: 𝑑 = 0.439 𝑁=144, 𝑘 = 3 𝛼=0.05 → 𝑑 𝑢 =1.774, 𝑑 𝐿 =1.693 𝛼=0.01→ 𝑑 𝑢 =1.665, 𝑑 𝐿 =1.584 𝐻 0 : no +ve autocorrelation, 𝐻 0 ∗ : no -ve autocorrelation The 𝑑 statistic is less than the critical 𝑑 𝐿 value ∴ we reject the null hypothesis ( 𝐻 0 ) of no +ve correlation at both 5% and 1% levels.

Question a) Limitations of Durbin-Watson statistic Lagged residuals only to first order Zones of indecision Not appropriate when lagged dependant variable is included The Breusch-Godfrey test allows: Higher order autocorrelation Still appropriate when a lagged dependant variable is included

Question b) Test for autocorrelation using the Breusch-Godfrey test statistic given by PC-GIVE. What (default) order of autocorrelation is being tested for here?

Question b) 𝑀𝑜𝑑𝑒𝑙→ 𝑇𝑒𝑠𝑡→ 𝑇𝑒𝑠𝑡…→ 𝐸𝑟𝑟𝑜𝑟 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑒𝑠𝑡 The order of the autocorrelation being tested here is 5th, e.g… 𝑢 𝑡 = 𝜌 1 𝑢 𝑡−1 + 𝜌 2 𝑢 𝑡−2 + 𝜌 3 𝑢 𝑡−3 + 𝜌 4 𝑢 𝑡−4 + 𝜌 5 𝑢 𝑡−5 + 𝜀 𝑡 Test statistic is 97.171~ 𝜒 5 2 𝜒 5 2 0.05 =11.0705, 𝜒 5 2 0.01 =15.0863 ∴ We reject 𝐻 0 (of no autocorrelation) at both 5% and 1% levels.

Question c) Construct the Breusch-Godfrey test for up to second order autocorrelation and test using the F statistic.

Breusch-Godfrey Test Method Estimate the model and save the residuals ( 𝑢 𝑡 ) Estimate: 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝛼 4 𝑢 𝑡−1 + 𝛼 5 𝑢 𝑡−2 + 𝜀 𝑡 Note the 𝑅 2 from Step 2 and calculate the 𝜒 2 test statistic as: 𝑁−𝑞 𝑅 2 ~ 𝜒 𝑞 2 Where: 𝑁 = 𝑓𝑢𝑙𝑙 𝑛𝑜. 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑞 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑎𝑔𝑠 (𝑜𝑟 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟 𝑜𝑓 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛) Compare the test statistic from Step 3 with the 𝜒 2 critical values at the 5% and 10% levels.

Breusch-Godfrey Test Construct the Breusch-Godfrey test, for second order autocorrelation, e.g. 𝑢 𝑡 = 𝜌 1 𝑢 𝑡−1 + 𝜌 2 𝑢 𝑡−2 𝐻 0 : 𝑛𝑜 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛, 𝜌 1 = 𝜌 2 =0 𝐻 1 : 𝐻 0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 Test Statistic: 𝑁−𝑞 𝑅 2 = 44−2 0.669224 95.0298~ 𝜒 2 2 𝜒 2 2 0.05 =5.991 𝜒 2 2 0.01 =9.210 ∴ We reject the null hypothesis of no autocorrelation at both 5% and 1% 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝛼 4 𝑢 𝑡−1 + 𝛼 5 𝑢 𝑡−2 + 𝜀 𝑡

Breusch-Godfrey Test (F-test) Method Estimate the model and save the residuals ( 𝑢 𝑡 ) Estimate two factor models: RES: 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝜀 𝑡 UNRES: 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝛼 4 𝑢 𝑡−1 + 𝛼 5 𝑢 𝑡−2 + 𝜀 𝑡 Over the same sample i.e. 𝑁−2 = 𝑁−𝑞 = 144−2 Undertake the F-test for: 𝐻 0 : 𝛼 4 = 𝛼 5 =0 𝐹= 𝑅 𝑈𝑁𝑅𝐸𝑆 2 − 𝑅 𝑅𝐸𝑆 2 𝑞 1− 𝑅 𝑈𝑁𝑅𝐸𝑆 2 𝑁 ∗ −𝐾 Where: 𝑞 =𝑛𝑜. 𝑜𝑓 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛𝑠 𝑘 = 𝑛𝑜. 𝑜𝑓 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑖𝑠 𝑈𝑁𝑅𝐸𝑆 𝑁 ∗ = 𝑛𝑜. 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑈𝑁𝑅𝐸𝑆 𝑠𝑎𝑚𝑝𝑙𝑒 (𝑁−𝑞)

Breusch-Godfrey Test (F-test) Construct the Breusch-Godfrey test, for second order autocorrelation. 𝐻 0 : 𝑛𝑜 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛, 𝛼 4 = 𝛼 5 =0 𝐻 1 : 𝐻 0 𝑖𝑠 𝑓𝑎𝑙𝑠𝑒 Test Statistic: 𝐹= 𝑅 𝑈𝑁𝑅𝐸𝑆 2 − 𝑅 𝑅𝐸𝑆 2 𝑞 1− 𝑅 𝑈𝑁𝑅𝐸𝑆 2 𝑁 ∗ −𝐾 𝐹= 0.669224−7.266996× 10 −6 2 1−0.669224 142−6 𝐹=137.57567~ 𝐹 𝑞, 𝑁 ∗ −𝐾 𝐹 2, 136 0.05 ≈3.07, 𝐹 2, 136 0.01 ≈4.79 ∴ We reject the null hypothesis of no autocorrelation at both 5% and 1% 𝑅𝐸𝑆: 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝜀 𝑡 𝑈𝑁𝑅𝐸𝑆: 𝑢 𝑡 = 𝛼 0 + 𝛼 1 log 𝐼 𝑡 + 𝛼 2 log 𝑊 𝑡 + 𝛼 3 𝑟 𝑡 + 𝛼 4 𝑢 𝑡−1 + 𝛼 5 𝑢 𝑡−2 + 𝜀 𝑡

Question c) 𝑀𝑜𝑑𝑒𝑙→ 𝑇𝑒𝑠𝑡→ 𝑇𝑒𝑠𝑡…→ 𝐸𝑟𝑟𝑜𝑟 𝑎𝑢𝑡𝑜𝑐𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑒𝑠𝑡 (′𝑡𝑜 𝑙𝑎𝑔′ = 2) Test Statistic: 96.262 ~ 𝜒 2 2 𝜒 2 2 0.05 =5.991 𝜒 2 2 0.01 =9.210 ∴ We reject the null hypothesis of no autocorrelation at both 5% and 1%

Question d) What are the consequences of your findings for the usefulness of the standard Ordinary Least Squares results for the consumption function above?

Question d) Consequences of autocorrelation OLS estimators are LUE but not BLUE (most efficient and unbiased) The estimated variances of OLS estimators are biased Usual 𝑡 and 𝐹 tests are unreliable The usual formula to compute the error variance is a biased estimator of the true 𝜎 2 The conventionally computed 𝑅 2 may be an unreliable measure of the true 𝑅 2