Applied Econometrics Applied Econometrics Second edition Dimitrios Asteriou and Stephen G. Hall Applied Econometrics: A Modern Approach using Eviews and Microfit © Dr D Asteriou Applied Econometrics AUTOCORRELATION 1. What is Autocorrelation 2. What Causes Autocorrelation 3. First and Higher Orders 4. Consequences of Autocorrelation 5. Detecting Autocorrelation 6. Resolving Autocorrelation 2 Applied Econometrics Learning Objectives 1. Understand the meaning of autocorrelation in the CLRM. 2. Find out what causes autocorrelation. 3. Distinguish among first and higher orders of autocorrelation. 4. Understand the consequences of autocorrelation on OLS estimates. 5. Detect autocorrelation through graph inspection. 6. Detect autocorrelation through formal econometric tests. 7. Distinguish among the wide range of available tests for detecting autocorrelation. 8. Perform autocorrelation tests using econometric software. 9. Resolve autocorrelation using econometric software. Applied Econometrics What is What is Autocorrelation Autocorrelation Assumption 6 of the CLRM states that the covariances and correlations between different disturbances are all zero: cov(ut, us)=0 for all t≠s This assumption states that the disturbances ut and us are independently distributed, which is called serial independence. Applied Econometrics What is Autocorrelation If this assumption is no longer valid, then the disturbances are not pairwise independent, but pairwise autocorrelated (or Serially Correlated). This means that an error occurring at period t may be carried over to the next period t+1. Autocorrelation is most likely to occur in time series data. In cross-sectional we can change the arrangement of the data without altering the results. Applied Econometrics What Causes Autocorrelation One factor that can cause autocorrelation is omitted variables. Suppose Yt is related to X2t and X3t, but we wrongfully do not include X3t in our model. The effect of X3t will be captured by the disturbances ut. If X3t like many economic series exhibit a trend over time, then X3t depends on X3t-1, X3t -2 and so on. Similarly then ut depends on ut-1, ut-2 and so on. Applied Econometrics What Causes Autocorrelation Another possible reason is misspecification. Suppose Yt is related to X2t with a quadratic relationship: Yt=β1+β2X22t+ut but we wrongfully assume and estimate a straight line: Yt=β1+β2X2t+ut Then the error term obtained from the straight line will depend on X22t. Applied Econometrics What Causes Autocorrelation A third reason is systematic errors in measurement. Suppose a company updates its inventory at a given period in time. If a systematic error occurred then the cumulative inventory stock will exhibit accumulated measurement errors. These errors will show up as an autocorrelated procedure Applied Econometrics First-Order Autocorrelation The simplest and most commonly observed is the first-order autocorrelation. Consider the multiple regression model: Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut in which the current observation of the error term ut is a function of the previous (lagged) observation of the error term: ut=ρut-1+et Applied Econometrics First-Order Autocorrelation The coefficient ρ is called the first-order autocorrelation coefficient and takes values fom -1 to +1. It is obvious that the size of ρ will determine the strength of serial correlation. We can have three different cases. Applied Econometrics First-Order Autocorrelation (a) If ρ is zero, then we have no autocorrelation. (b) If ρ approaches unity, the value of the previous observation of the error becomes more important in determining the value of the current error and therefore high degree of autocorrelation exists. In this case we have positive autocorrelation. (c) If ρ approaches -1, we have high degree of negative autocorrelation. Applied Econometrics First-Order Autocorrelation Applied Econometrics First-Order Autocorrelation Applied Econometrics Higher-Order Autocorrelation Second-order when: ut=ρ1ut-1+ ρ2ut-2+et Third-order when ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +et p-th order when: ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et Applied Econometrics Consequences of Autocorrelation 1. The OLS estimators are still unbiased and consistent. This is because both unbiasedness and consistency do not depend on assumption 6 which is in this case violated. 2. The OLS estimators will be inefficient and therefore no longer BLUE. 3. The estimated variances of the regression coefficients will be biased and inconsistent, and therefore hypothesis testing is no longer valid. In most of the cases, the R2 will be overestimated and the t-statistics will tend to be higher. Applied Econometrics Detecting Autocorrelation There are two ways in general. The first is the informal way which is done through graphs and therefore we call it the graphical method. The second is through formal tests for autocorrelation, like the following ones: 1. The Durbin Watson Test 2. The Breusch-Godfrey Test 3. The Durbin’s h Test (for the presence of lagged dependent variables) 4. The Engle’s ARCH Test Applied Econometrics Detecting Autocorrelation We have the following series (quarterly data from 1985q1 to 1994q2): lcons = the consumer’s expenditure on food ldisp = disposable income lprice = the relative price index of food Typing in Eviews the following command ls lcons c ldisp lprice we get the regression results. Applied Econometrics Detecting Autocorrelation Then we can store the residuals of this regression in a vector by typing the command: genr res01=resid And a plot of the residuals can be obtained by: plot res01 While a scatter of the residuals against their lagged terms can be obtained by: scat res01(-1) res01 Applied Econometrics Detecting Autocorrelation .12 .08 .04 .00 -.04 -.08 85 86 87 88 89 90 RES01 91 92 93 Applied Econometrics Detecting Autocorrelation .12 RES01 .08 .04 .00 -.04 -.08 -.08 -.04 .00 .04 RES01(-1) .08 .12 Applied Econometrics The Durbin Watson Test The following assumptions should be satisfied: 1. The regression model includes a constant 2. Autocorrelation is assumed to be of firstorder only 3. The equation does not include a lagged dependent variable as an explanatory variable Applied Econometrics The Durbin Watson Test Step 1: Estimate the model by OLS and obtain the residuals Step 2: Calculate the DW statistic Step 3: Construct the table with the calculated DW statistic and the dU, dL, 4-dU and 4-dL critical values. Step 4: Conclude Applied Econometrics The Durbin Watson Test Zone of No indecision autocorrelation +ve autoc 0 dL dU 2 Zone of indecision 4-dU -ve autoc 4-dL 4 Applied Econometrics The Durbin Watson Test Drawbacks of the DW test 1. It may give inconclusive results 2. It is not applicable when a lagged dependent variable is used 3. It can’t take into account higher order of autocorrelation Applied Econometrics The Breusch-Godfrey Test It is a Lagrange Multiplier Test that resolves the drawbacks of the DW test. Consider the model: Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut where: ut=ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et Applied Econometrics The Breusch-Godfrey Test Combining those two we get: Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ +ρ1ut-1+ ρ2ut-2+ρ3ut-3 +…+ ρput-p +et The null and the alternative hypotheses are: H0: ρ1= ρ2=…= ρp=0 no autocorrelation Ha: at least one of the ρ’s is not zero, thus, autocorrelation Applied Econometrics The Breusch-Godfrey Test Step 1: Estimate the model and obtain the residuals Step 2: Run the full LM model with the number of lags used being determined by the assumed order of autocorrelation. Step 3: Compute the LM statistic = (n-ρ)R2 from the LM model and compare it with the chisquare critical value. Step 4: Conclude Applied Econometrics The Durbin’s Durbin’s hh Test The Test When there are lagged dependent variables (i.e. Yt-1) then the DW test is not applicable. Durbin developed an alternative test statistic, named the hstatistic, which is calculated by: n DW h 1 2 2 1 n ˆ Where sigma of gamma hat square is the variance of the estimated coefficient of the lagged dependent variable. This statistic is distributed following the normal distribution Applied Econometrics The Durbin’s h Test Dependent Variable: LOG(CONS) Included observations: 37 after adjustments Variable C LOG(INC) LOG(CPI) LOG(CONS(-1)) Coefficient 0.834242 0.227634 -0.259918 0.854041 R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat 0.940878 0.935503 0.028001 0.025874 81.91016 1.658128 Std. Error 0.626564 0.188911 0.110072 0.089494 Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic) t-Statistic 1.331456 1.204981 -2.361344 9.542982 Prob. 0.1922 0.2368 0.0243 0.0000 4.582683 0.110256 -4.211360 -4.037207 175.0558 0.000000 Applied Econometrics The Durbin’s h Test n DW h 1 2 2 1 n ˆ 37 1.658 1 1.2971 2 2 1 37 * 0.089 Applied Econometrics Resolving Autocorrelation We have two different cases: (a) When ρ is known (b) When ρ is unknown Applied Econometrics Resolving Autocorrelation (when ρ is known) Consider the model Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut where ut=ρ1ut-1+et Applied Econometrics Resolving Autocorrelation (when ρ is known) Write the model of t-1: Yt-1=β1+β2X2t-1+β3X3t-1+β4X4t-1+…+βkXkt-1+ut-1 Multiply both sides by ρ to get ρYt-1= ρβ1+ ρβ2X2t-1+ ρβ3X3t-1+ ρβ4X4t-1 +…+ ρ βkXkt-1+ ρut-1 Applied Econometrics Resolving Autocorrelation (when ρ is known) Subtract those two equations: Yt-ρYt-1= (1-ρ)β1+ β2(X2t-ρX2t-1)+ β3(X3t-ρX3t-1)+ +…+ βk(Xkt-ρXkt-1)+(ut-ρut-1) or Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+et Where now the problem of autocorrelation is resolved because et is no longer autocorrelated. Applied Econometrics Resolving Autocorrelation (when ρ is known) Note that because from the transformation we lose one observation, in order to avoid that loss we generate Y1 and Xi1 as follows: Y*1=Y1 sqrt(1- ρ2) X*i1=Xi1 sqrt(1-ρ2) This transformation is known as the quasidifferencing or generalised differencing. Applied Econometrics Resolving Autocorrelation (when ρ is unknown)) The Cochrane-Orcutt iterative procedure. Step 1: Estimate the regression and obtain residuals Step 2: Estimate ρ from regressing the residuals to its lagged terms. Step 3: Transform the original variables as starred variables using the obtained from step 2. Step 4: Run the regression again with the transformed variables and obtain residuals. Step 5 and on: Continue repeating steps 2 to 4 for several rounds until (stopping rule) the estimates of from two successive iterations differ by no more than some preselected small value, such as 0.001.