WHAT IF ASSUMPTIONS ARE INVALID? Chapter 5, (selected) - S&W Dr Tu Thuy Anh Faculty of International Economics BASIC ASSUMPTIONS E(ui) = Var(ui) = σ2 cov(ui, uj) = for i #j ui~ N(0, σ2) No perfect collinearity among independent vars Xj: non random, Y: random Assumptions     Multicollinearity (p 201-203, 204-207 S&W) Normality Heteroscedasticity (p158-164; 178-180 S&W) Autocorrelation HIGH COLINEARITY- EXAMPLE  Recall: perfect co-linearity  => what if collinearity among independent variables is high?  Examples:  price of beef/ pork  K and L of a firm  VNindex and HSTC index  Money supply and GDP  If there are independent variables: r23 ~+/- HIGH COLINEARITY- CONSEQUENCE      large variance of estimates wide confidence interval =>? small tob => less chance to reject H0 “wrong” sign of the estimates however, still unbiased estimates only a problem if consequences are serious HIGH COLINEARITY- SYMPTOMS  Example:  Exp^ = 1.5 - 0.2Income +0.1Wealth Wrong sign  Variable C P PA Coefficient Std Error t-Statistic Prob 1207.06 1575.06 0.77 0.45 -146.90 479.15 -0.31 0.76 0.15 6.34 0.02 0.98 R2 =0.91 R2 large, but few significant ratio HIGH COLINEARITY- DETECTION  Run an auxiliary model: Dependent variable: P Variable Coefficient Std Error t-Statistic Prob C 3.28 0.05 61.35 0.00 PA 0.01 0.00 38.19 0.00 R-squared 0.99 Mean dependent var 5.24 Log likelihood 26.27 F-statistic 1458.58 Durbin-Watson stat 1.27 Prob(F-statistic) 0.00  If R2 is large?  if VIF>10? VIF = 1/(1-R2)) HIGH COLINEARITY- CAUSE/ CURE  Cause:  lack of data  nature of variables of interest  Cure:  Collect more data  Use other source of information: CRS, …  Delete one or more variables  Transform the variables: taking log, ratio, growth NORMALITY- CONSEQUENCES  ui~ N(0, σ2)  If this assumption does not hold, then estimates are still unbiased, but we will not be able to assess which parameters are significant - the normality of the estimates will not hold - the significance tests will not follow a t-Student distribution - and the joint significance test will not follow an F distribution Histogram 10 NORMALITY- DETECTION The Jarque-Bera test: JB = N/6 (S2 + K2 /4) where S and K are the sample Skewness and Kurtosis statistics The JB test has an asymptotic chi-square distribution with two degrees of freedom 11 NORMALITY- CAUSE/CURE  Nature of data  Use the Central Limit Theorem to assure that even that u (hence Y) does not come from a normal distribution, the parameters estimates will be asymptotically normal and consequently we will be able to perform the usual inference  But, how large has to be the sample size to achieve normality in the parameter estimates? - Some econometricians propose N>30, but it can be not enough in some situations - The quality of the approximation depends not only on N, but also on N-K, the degrees of freedom 12 HETEROSCEDASTICITY- CONSEQUENCES  What does it mean: var(ui) = σ2i  OLS Estimates are still unbiased  Biased estimation of var(aˆ j )  => Confidence Intervals are invalid  => Invalid t, F test Need to be cured 13 HETEROSCEDASTICITY- DETECTION  White test:  H0 : Var(ui) = σ2 for all i  Regress the original model =>obtain ei  Run (with cross-term): e  1   X   X   X 22   X 32   X X  u  If nR (1)   2 (k  1) => R2(1) ( R (1)) / (k  1) F  f (k  1, n  k ) (1  R (1)) / (n  k ) k: n0 of coeffs in model Reject H0  Do the same with “no cross term” 14 HETEROSCEDASTICITY- CAUSE/CURE  Causes:  Nature of data  Wrong functional form  etc  Cure:  Use “option Robust standard error” in the software  Transform variables depending on the form of heteroscedasticity  Example: if σ2i = aK2i  => dividing both sides of the model by K 15 AUTOCORRELATION  If the assumption does not hold: cov(ui; uj) # for i # j  Form of autocorrelation:  ut = ρut-1 + vt =>AR(1);  v(t): random error, satisfies assumptions 1-4  If ρ >0: positive autocorrelation  If ρ AR(p) 16 AUTOCORRELATION -CONSEQUENCES  Consequence:  OLS Estimates are still unbiased  Biased estimation of var(aˆ j ) => invalid Confidence Interval  Invalid t, F test  Biased estimation of ˆ NEED TO BE CURED 17 AUTOCORRELATION - DETECTION + _ no dL dU 4-dU 4-dL NO CONCLUSION  Durbin Watson test, can be used in situations:  AR (1)  No lag value of the dependent variable in the model  No missing observation 18 AUTOCORRELATION - DETECTION  B-G test: et = a1 + a2 Xt + ρ1et-1+ + ρp et-p +vt => R2(1) => R2(2) et = a1 + a2 Xt + vt If: 2 nR (1)    ( p) or (R2(1)R2(2))/ p F  f(p,nk*) (1R (1))/(nk*) Autocorrelation of order p 19 AUTOCORRELATION - CURE  AR(1): ut = ρut-1 +vt  Estimate ρ, then run GLS as follows:  Set:  Y* = Y – ρ’Y(-1);  X* = X – ρ’X(-1)  Run OLS for the following regression: Y* = β1+ β2X* + v 20 ...  Example:  Exp^ = 1 .5 - 0.2Income +0.1Wealth Wrong sign  Variable C P PA Coefficient Std Error t-Statistic Prob 1207.06 157 5.06 0.77 0. 45 -146.90 479. 15 -0.31 0.76 0. 15 6.34 0.02 0.98 R2 =0.91... Coefficient Std Error t-Statistic Prob C 3.28 0. 05 61. 35 0.00 PA 0.01 0.00 38.19 0.00 R-squared 0.99 Mean dependent var 5. 24 Log likelihood 26.27 F-statistic 1 458 .58 Durbin-Watson stat 1.27 Prob(F-statistic)...BASIC ASSUMPTIONS E(ui) = Var(ui) = σ2 cov(ui, uj) = for i #j ui~ N(0, σ2) No perfect collinearity among independent vars Xj: non random, Y: random Assumptions     Multicollinearity