Some Useful Econometric Techniques

1. Some Useful Econometric Techniques Selcuk Caner

2. Outline • Descriptive Statistics • Ordinary Least Squares • Regression Tests and Statistics • Violation of Assumptions in OLS Estimation • Multicollinearity • Heteroscedasticity • Autocorrelation • Specification Errors • Forecasting • Unit Roots, Spurious Regressions, and cointegration

3. Descriptive Statistics • Useful estimators summarizing the probability distribution of a variable: • Mean • Standard Deviation

4. Descriptive Statistics (Cont.) • Skewness (symmetry) • Kurtosis (thickness)

5. Ordinary Least Squares (OLS) • Estimation • Model • The OLS requires: • Linear relationship between Y and X, • X is nonstochastic, • E(et) = 0 , Var(et) = s2 and Cov(et, es)=0 for t not equal to s.

6. Ordinary Least Squares (OLS) (Cont.) • The OLS estimator for b0 and b1 are found by minimizing the sum of squared errors (SSE):

7. Ordinary Least Squares (OLS) (Cont.) o o o o o o o o X Xt

8. Ordinary Least Squares (OLS) (Cont.) • Minimizing the SSE is equivalent to: • Estimators are:

9. Ordinary Least Squares (OLS) (Cont.) • Properties of OLS estimators: • They are normally distributed • Minimum variance and unbiased estimators

10. Ordinary Least Squares (OLS) (Cont.) Multiple regression, in matrix form, Y= Tx1 vector of dependent variables X = Txk matrix of independent variables (first column is all ones) b= kx1 vector of unknown parameters e = Tx1 vector of error terms

11. Ordinary Least Squares (OLS) (Cont.) Estimator of the multiple regression model: • X’X is the variance-covariance matrix of the components of X. • X’Y is the vector of covariances between X and Y. • It is an unbiased estimator and normally distributed

12. Example: Private Investment • FIRt = b0 + b1RINT t-1 + b2INFL t-1 + b3RGDP t-1 + b4NKFLOW t-1 + et • One can run this regression to estimate private fixed investment • A negative function of real interest rates (RINT) • A negative function of inflation (INFL) • A positive function of real GDP (RGDP) • A positive function of net capital flows (NKFLOW)

13. Regression Statistics and Tests • R2 is the measure if goodness of fit: • Limitations: • Depends on the assumption that the model is correctly specified • R2 is sensitive to the number of independent variables • If intercept is constrained to be equal to zero, then R2 may be negative.

14. Meaning of R2 o o o o o o o o X Xt

15. Regression Statistics and Tests • Adjusted R2 to overcome limitations of • R2 = 1-SSE/(T- K)/TSS/(T-1) • Is bi statistically different from zero? • When et is normally distributed, use t-statistic to test the null hypothesis bi = 0. • A simple rule: if t(T-k) > 2 then bi is significant.

16. Regression Statistics and Tests • Testing the model: • F-test: F-statistics with k-1 and T-k degrees of freedom is used to test for the null hypothesis: • b1=b2=b3=…=b k=0 • The f-statistics is: • The F test may allow the null hypothesis b1=b2=b3=…=b k=0 to be rejected even when none of the coefficients are statistically significant by individual t-tests.

17. Violations of OLS Assumptions • Multicollinearity • When 2 or more variables are correlated (in the multi variable case) with each other. E.g., • Result: high standard errors for the parameters and statistically insignificant coefficients. • Indications: • Relatively high correlations between one or more explanatory variables. • High R2 with few significant t-statistics. Why?

18. Violations of OLS Assumptions (Cont.)

19. Violations of OLS Assumptions (Cont.) • Heteroscedasticity: when error terms do not have constant variances s2. • Consequences for the OLS estimators: • They are unbiased [E(b)=b] but not efficient. Their variances are not the minimum variance. • Test: White’s heteroscedasticty test. • If there are ARCH effects, use the GARCH models to account for volatility clustering effects.

20. Violations of OLS Assumptions (Cont.) • Autocorrelation: when the error terms from different time periods are correlated [et=f(et-1,et-2,…)]: • Consequences for the OLS estimators: • They are unbiased [E(b)=b] but not efficient. • Test for serial correlation: Durbin-Watson for first order serial correlation:

21. Violations of OLS Assumptions (Cont.) • Autocorrelation (cont.): • Test for serial correlation (cont.) • Durbin-Watson statistic (cont.) • The DW statistic is approximately equal to: where • Note, if r1=1 then DW =0. If r1=-1 then DW =4. For r1=0, DW =2. • Ljung-Box Q test statistic for higher order correlation.

22. Specification Errors • Omitted variables: • True model: • Regression model: • Then, the estimator for b1 is biased.

23. Specification Errors (Cont.) • Irrelevant variables: • True model: • Regression model: • Then, the estimator for b1 is still unbiased. Only efficiency declines, since the variance of b1* will be larger than the variance of b1.

24. A Naïve Estimation • Estimate aggregate demand elasticity e • Using historical consumption data: • Estimate the regression equation: • ln(QDt) = a + b*ln(Pt) • b is an estimate of e • Forecast change in consumption price, DP • Estimate change in demand as: • (DQD/QD)F = e * (DP/P)F

25. A Regression Result

26. Stationarity (ADF Test)

27. Error Correction Model (ECM) for Non-Stationarity • One can try regression of first differences. • However, first differences do not use information on levels. • It mixes long-term relationship with the short-term changes. • Error correction model (ECM) can separate long-term and short-term relationships.

28. Results of ECM

29. Interpretation of the Estimated Regression • ln QDt – lnQD t-1 = -5.327-0.348*(lnCPIt – lnCPI t-1) – 0.697* (lnQD t-1- 1.267 lnCPI t-1) Short-run Effect Long-run Effect Error Correction Coefficient

30. Forecasting • A forecast is: • A quantitative estimate about the likelihood of future events which is developed on the basis of current and past information. • Some useful definitions: • Point forecast: predicts a single number for Y in each forecast period • Interval forecast: indicates an interval in which the realized value of Y will lie.

31. Unconditional Forecasting • First estimate the econometric model • Then, compute: assuming XT+1 is known. This is the point forecast.

32. Unconditional Forecasting (Cont.) • The forecast error is: • The 95% confidence interval for YT+1 is: • where • Which provides a good measure of the precision of the forecast.

33. Conditional Forecasting • If XT+1 is not known and needs to be forecasted. • The stochastic nature of the predicted values for Xs leads to forecasts that are less reliable. • The forecasted value of Y at time T+1 is

34. Unit Roots, Spurious Regressions, and Cointegration • Simulate the processes • where et ~N(0,4) and • where ut ~N(0,9).

35. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • Spurious regressions: • Granger and Newbold(1974) demonstrated that macroeconomic variable data are trended upwards and that in regressions involving the levels of such data, the standard significance tests are misleading. The conventional t and F tests reject the hypothesis of no relationship when in fact there might be one. • Symptom: R2 > DW is a good rule of thumb to suspect that the estimated regression is spurious.

36. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • Unit roots: • If a variable behaves like • Then its variance will be infinite since, • This is a non-stationary variable. E.g., where et ~N(0,4). This would result with a forever increasing series.

37. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • The series can be made stationary by taking first difference of Yt, • The series has finite variance and is a stationary variable. The original series Yt is said to be integrated of order one [I(1)].

38. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • A trend-stationary variable also has a finite variance. • The process is non-stationary and does not have a finite variance.

39. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • But the variable, • Is stationary and has a finite variance if abs(r)<1. E.g., where et ~N(0,4).

40. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • Tests for unit roots: Dickey-Fuller Test • Case of I(1) • Null hypothesis: • Alternative hypothesis: • Run regression: • And test (r-1)=0 by comparing the t-statistic with MacKinnon critical values for rejection of the hypothesis of a unit root.

41. Unit Roots, Spurious Regressions, and Cointegration (Cont.) Case of Random Walk (RW) • Null hypothesis: • Alternative hypothesis: • Run regression: • And test (r-1)=0 by comparing the t-statistic with MacKinnon critical values for rejection of the hypothesis of a unit root.

42. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • DF tests on macroeconomic variables: • Most macroeconomic flows and stocks related to the population size such as output, consumption or employment are I(1) while price levels are I(2). E.g., GDP is I(1) while interest rates are I(2).

43. Unit Roots, Spurious Regressions, and Cointegration (Cont.) • Cointegration • If two series are both I(1), there may be a b1 such that • Is I(0). The implication is that the two series are drifting upward together at roughly the same rate. • Two series satisfying the above requirement are said to be cointegrated and the vector is a cointegrating vector.