Oct 26 2006 LEC 9 ECON 240A 1 L Phillips Experimental Method Clinical Trials and Experimental Design I Introduction Economists 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 know which group they are in Oct 26 2006 LEC 9 ECON 240A 2 L Phillips Experimental Method Clinical Trials and Experimental Design For 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 wifebeater 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 frequency in 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 Squares In 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 Phillips Experimental Method Clinical Trials and Experimental Design We 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 of the 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 b b 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 Squares Violations of the assumptions underlying the properties of least squares estimators lead 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 Phillips Experimental Method Clinical Trials and Experimental Design For 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 a and b 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 Squares We 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 are distributed randomly as assumed or whether we can detect patterns Patterns can reveal outliers such as very large errors These may be due to a data error for example caused by a mistake in copying the data Figure 1 Fitted Vs Actual Net Returns UC Stock Index Fund and Residuals by Month 10 5 0 4 5 2 0 2 4 99 09 99 11 Residual 00 01 00 03 Actual 00 05 Fitted 00 07 Oct 26 2006 LEC 9 ECON 240A 5 L Phillips Experimental Method Clinical Trials and Experimental Design With time series data such as our sequence of twelve months of net rates of return for the UC stock index fund patterns in Figure 1 may be revealing For example the estimated residual is within plus or minus two standard deviations of zero for most
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