I. IntroductionII. The Assumptions of Least SquaresIII. The Pathologies of Least SquaresIV. Graphical Diagnostics for Least SquaresV. The Mean and Variance of the OLS Estimate for the InterceptVI. Testing Hypotheses About the InterceptVII. The Expected Value of the Sum of Squared ResidualsVIII. Trend Analysis and Interpreting Parameter EstimatesIX. Bivariate RelationshipsX. Clinical Trials: Comparing Success(Failure) Rates for Experimentals and ControlsVIII. Experimental DesignCabBrand ABrand BDifferenceOct. 26, 2006 LEC #9 ECON 240A-1 L. PhillipsExperimental Method, Clinical Trials and Experimental DesignI. IntroductionEconomists usually rely on regression to estimate the effect of one variable on another. An example is the effect of deterrence variables, such as the probability of arrest (measured as the ratio of arrests to reported offenses), x, on the felony offense rate, y. A critique of this approach is that one should control for the causes of crime, indicated symbolically here by the variable w. But, as the critique continues, not enough is known about the causes of crime to appropriately measure w. A proxy variable, q, included to control for causality, may not be adequate for the purpose. So, any estimated deterrence effect is suspect because the variation in the offense rate due to causal factors has not been controlled for statistically in a convincing manner. In summary, one should estimate:y = a + b x + c w + u, (1)but estimates,y = a’ + b’ x + c’ q + u’. (2)An alternative approach, used frequently in some other disciplines, is the experimental method. The key is to randomly assign subjects into two groups, the experimental group and the control group. This approach has been used to study the deterrent effect of punishment in situations of domestic violence.The idea is that you can study the “response” of a subject to a “stimulus” or treatment, and compare the response of individuals in the experimental group, who receive the treatment, to the response of individuals in the control group, who do not receive the treatment. This experiment is conducted “blind”, i.e. the subjects do not knowwhich group they are in.Oct. 26, 2006 LEC #9 ECON 240A-2 L. PhillipsExperimental Method, Clinical Trials and Experimental DesignFor example, in the case of domestic violence, a call to 911 for help is always answered by the police, who go to the home and try and calm the occupants. In an experiment however, the case is assigned at random to the experimental group or the control group. If it is assigned to the experimental group, the officers may arrest the wife-beater, for example and haul him off for a night in jail. This punishment has been found to reduce, i.e. deter, the frequency of future domestic violence, compared to the frequencyin the control group. The point about random assignment to the experimental or control group is that it is a mechanism for dealing with the unknown causes of domestic violence. The random assignment of subjects should insure that there are as many violent wife-beaters, or heavies, in the control group as in the experimental group, thereby permitting isolation of the deterrent effect of punishment, i.e. arrest and jail. The experimental method and stimulus-response studies are widely used in many fields for numerous applications. These include the testing of drugs, pesticides, fertilizers etc. Before looking at the experimental method, and testing for differences in proportions or sample means between the experimental and control groups, we will tie up some loose ends for least squares.II. The Assumptions of Least SquaresIn Lecture Eight, we used a number of assumptions in investigating the formulae for least squares estimates. We will summarize them here.#1. The expected value of the error is zero, E(u) = 0.We used this assumption in Lecture Eight, p.1.#2. The error is independent of the explanatory variable, E{[x – Ex] u} =0.Oct. 26, 2006 LEC #9 ECON 240A-3 L. PhillipsExperimental Method, Clinical Trials and Experimental DesignWe used this assumption on p.1 and p.3 of Lecture Eight.#3. The errors are independent of one another, E[u(i) u(j)] =0. We used this assumption on p. 9 of Lecture Eight.#4. The variance is the same, i.e homoskedastic, for all of the errors, E[u(I)]2 = 2 We used this on p.9 of Lecture Eight.#5. The error is distributed normally, with mean zero and variance 2 .This is the reason the estimate of the unexplained mean square, i.e. the estimate ofthe variance of the error, is distributed Chi-Square with n-2 degrees of freedom.In turn, this is the basis for the distributional assumptions underlying Student’s t-test and the F-test. The estimated slope and intercept are distributed normally if the errors are normal, and a variable such as )ˆ(ˆ/ˆbb is the ratio of a normal variable to the square root of a Chi-Square variable, and hence has the t distribution.III. The Pathologies of Least SquaresViolations of the assumptions underlying the properties of least squares estimatorslead to various pathologies or problems that need remedies. For example, if assumption four is false, then the error is said to be heteroskedastic. The consequence is that the ordinary least squares estimators are no longer best in the sense of being minimum variance estimators or using the information in the data efficiently. There are tests developed for these econometric problems as well as various remedies, and you will study these techniques of analysis in Econ 240B.Oct. 26, 2006 LEC #9 ECON 240A-4 L. PhillipsExperimental Method, Clinical Trials and Experimental DesignFor regressions using time series, the errors may be correlated violating assumption three. This is called autocorrelation. If the explanatory variable is not independent of the error, then the OLS parameter estimates aˆ and bˆ are biased and inconsistent, i.e the uncertainty or variability in these estimators no longer diminishes as sample size grows.IV. Graphical Diagnostics for Least SquaresWe are emphasizing exploratory analysis and graphical methods. One diagnostic tool that we will resort to is an examination of the actual and fitted data, as displayed in Figure 4 of Lecture Eight. This may reveal observations that are outliers, i.e. lie far from the fitted relationship. Another useful visual tool is a plot of the estimated residuals, as shown for that regression in Figure 1, below. Looking at such a plot, we check to see if the estimated residuals
View Full Document
Unlocking...