Econometrics Part Three

Transcription

1 !1 I. Heteroskedasticity A. Definition 1. The variance of the error term is correlated with one of the explanatory variables 2. Example -- the variance of actual spending around the consumption line increases as income increases Consumption a)! B. Effect 1. The b's are still unbiased, but the standard errors (the estimated standard deviation for each b, or Sb) are biased (the reported value is higher or lower than the actual value). a) With positive heteroskedasticity (variance increases with X, as in the example above), the reported standard error is low, increasing the t- statistic. This may cause you to reject the hypothesis that β = 0 when it is true. b) The opposite is true with negative heteroskedasticity. C. Testing for heteroskedasticity Income 1. Graph the regression residuals on the Y axis each independent variable on the X axis and look for differences in the amount of variation in the residuals. If there is no heteroskedasticity, it will look like this: Residuals 0 Income a) b) If there is a small number of cases at one end of the X axis, the amount of variation will appear smaller because the probability of a large or small

2 !2 II. D. Correcting value is low in a small sample. This is not strong evidence of heteroskedasticity. 1. One possible solution is to change the dependent variable from a total to a rate. Instead of explaining the amount spent at different levels of income, explain the percentage of income spent at different levels of income. Non-normal errors A. OLS is based on the assumption that the errors are normally distributed. This means that most of the errors are close to zero, that is, most of the data points are close to the actual line. This is the rationale for estimating the line as close as possible to the data points. B. The test for statistical significance for each coefficient is based on the assumption of normally distributed errors. The estimated coefficients have a t distribution only if the errors have a normal distribution. If the errors are not normal, then the t statistic does not correctly estimate the probability that the variable has no effect. C. To test for normality of the errors, SPSS can plot the histogram for the residuals. If it is approximately normal, then OLS is appropriate to use. III. Simultaneous Equations A. Example -- supply and demand 1. Simple model a) S: Q = a + b P + ε b) D: Q = c - d P + µ 2. In this case, errors (ε and µ) represent random shifts of the supply and demand curve, respectively. Thus, the observations would be a 'cloud' of points around the stable equilibrium (where a + bp = c - dp). Fitting a regression line to these points will estimate neither the demand or supply curves. 3. If errors are measured vertically (in $s), then the estimated line will be horizontal). If the errors are measured horizontally (in units of the product, as in the two equations above), then the estimated line will be vertical. P estimated line S D a)! Q

3 !3 4. To estimate the supply curve, you need something (like a change in income) to shift the demand curve. a) It is important that whatever shifts the demand curve does not also shift the supply curve. P S estimated line D D' b)! 5. More complex model a) S: Q = a + b P + c P r + ε b) D: Q = d - e P + f I + µ 6. Able to estimate both S and D equations a) S is identified because it does not include I (to shift D) b) D is identified because it does not include P r (to shift S) B. In general, an equation is identified if it does not include the same variables that shift the other equations. 1. Rule is that the number of excluded variables in an equation is at least as big as the number of equations If there are two equations, that means that the supply equation is identified because it does not contain the I variable, while the demand equation is identified because it does not contain the P r variable. IV. Autocorrelation A. Definition Q 1. In general, autocorrelation occurs when a variable from period t is correlated with the same variable from period t-n. Most common is first degree autocorrelation, where Z t is correlated with Z t-1 (Z in the previous period). a) Positive autocorrelation = if Z t-1 is large then Z t will also be large (same sign). b) Negative autocorrelation = if Z t-1 is large and positive then Z t will be large and negative. 2. The problem with OLS occurs when the error term (ε) is autocorrelated. a) For example ε t = ρ ε t-1 + µ t, which means that a certain fraction (ρ) of the error in the previous period (ε t-1 ) carries over to the current error (ε t ), plus an additional random effect (µ t ).

4 !4 B. Using OLS when the errors are autocorrelated results in 1. unbiased estimated coefficients, but 2. inefficiency = the estimated coefficients are less likely to be close to the actual coefficients (bigger standard deviations), and 3. biased standard errors = the reported standard deviation of each estimated coefficient is likely to be too high or too low, depending on the type of autocorrelation. C. Cause a) With positive autocorrelation, the reported standard errors are understated, resulting in overly large t statistics (so may reject null hypothesis when it is true). (1) Don't ask why -- there is no obvious reason! b) With negative autocorrelation, the standard errors are overstated, resulting in too small t statistics. 1. Usually due to some form of inertia on the part of the dependent variable = pure ac. If it above the predicted value in one period, it will probably still be above the predicted value next period. 2. May be due to the omission of an independent variable that is itself autocorrelated = impure ac. This is even worse than simple autocorrelation, since the coefficient estimates will also be biased and inconsistent. D. Testing for autocorrelation 1. Have the computer calculate the Durbin-Watson statistic a) If there is no autocorrelation, then d 2. b) For perfect positive autocorrelation (identical errors over time), then d 0. c) For perfect negative autocorrelation (errors have same magnitude but opposite signs), then d Test the null hypothesis Ho: no positive autocorrelation against the alternative hypothesis of positive autocorrelation. a) The idea is to reject the hypothesis if d is close to 0 and accept it if d is close to 2. To do this, we need to (1) decide what close means and (2) decide what to do if d is not close to 0 or 2 (like d=1). (1) For specific n and k, the Durbin-Watson tables give d u and d L for a given level of significance (usually 5% or 1%) (2) If d u < d < 2, then accept the null hypothesis (since d is "close" to 2). (3) If d < d L, then reject the null hypothesis and accept alternative hypothesis of positive autocorrelation. (4) If d L < d < d u, then the test is inconclusive

5 !5 3. If d > 2, so suspect negative autocorrelation, test Ho: no negative autocorrelation using 4 d u as the lower bound and 4 - d L as the upper bound (the mirror images of d u and d L around 2) 4. Warning: d is not appropriate if lagged dependent variable is used as an explanatory variable. E. Example 1. Suppose that d = 1.65 when n=50 and k=3. Using the 5% significance level, d L = and d u = from the DW tables. Since d is between those values, the test is inconclusive. If it had been smaller than 1.421, then would conclude that there is positive autocorrelation with 95% certainty. If it was bigger than 1.674, would conclude that there is no autocorrelation, again with 95% certainty.