Two-Variable Regression: Interval Estimation and Hypothesis Testing

Transcription

1 Two-Variable Regression: Interval Estimation and Hypothesis Testing Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

2 Objectives By the end of this meeting, participants should be able to: Calculate confidence intervals for regression coefficients and the standard error of regression. Conduct hypothesis tests for regression coefficients and the standard error of regression. Conduct an analysis of variance (ANOVA) for a regression model. Make predictions about the mean of Y or individual values of Y, including measures of uncertainty. Test whether the residuals of an estimated model are normally distributed. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

3 Interpreting a Confidence Interval α =Type I error rate. What is a Type II error? Which of these is the correct interpretation of a (1 α) confidence interval? An interval that has a 1 α% chance of containing the true value of the parameter. An interval that over 1 α% of replications contains the true value of the parameter, on average. Note: If you use Bayesian methods, you can make different kinds of statements. Confidence Intervals Coverage Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

4 Why the t Distribution is Critical for Inference From the sampling distribution of ˆβ 2, we know that Z = ˆβ 2 β 2 distributed Z N (0, 1). This can be re-written as: Z = ( ˆβ 2 β 2 ) (Xi X ) 2 standard error of regression. σ σ ˆβ 2 for σ the Since we don t know σ, we often must substitute our estimate ˆσ, yielding t = ( ˆβ 2 β 2 ) (Xi X ) 2. ˆσ The ratio of a normally distributed variable to a χ 2 variable with n 2 degrees of freedom has a t distribution with n 2 degrees of freedom. is Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

5 Confidence Intervals for Regression Coefficients Therefore, for the right t distribution, we can use the following fact: [ ] Pr t α/2 ˆβ 2 β 2 ˆσ ˆβ 2 t α/2 = 1 α Doing some algebra yields: [ Pr ˆβ2 t α/2ˆσ β ˆβ2 2 ˆβ 2 + t α/2ˆσ ˆβ2] = 1 α Thus, our 100(1 α) percent confidence interval for β 2 is ˆβ 2 ± t α/2ˆσ ˆβ2. By a similar argument, our 100(1 α) percent confidence interval for β 1 is ˆβ 1 ± t α/2ˆσ ˆβ1. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

6 Confidence Intervals for Error Variance of Regression Recall: χ 2 = (n 2) ˆσ2. σ 2 This variable is distributed chi-squared with n 2 degrees of freedom. Therefore we know this fact: [ ] Pr χ 2 1 α/2 χ2 χ 2 α/2 = 1 α Doing some algebra yields: [ Pr (n 2) ˆσ2 χ 2 σ 2 (n 2) α/2 ˆσ 2 χ 2 1 α/2 ] = 1 α which is the simplest way to describe our confidence interval for σ 2. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

7 Hypothesis Testing Presumably we estimate a regression model to test some theory. We would like to make a statement about whether our theoretical relationship holds in the larger population. Hence, hypothesis tests about population slope coefficients are conducted daily in political science and are a major workhorse in our research. By default, software conducts the following hypothesis test: H 0 : β 2 = 0 H 1 : β 2 0 H 0 is the null hypothesis, and H 1 (also called H A ) is the alternative hypothesis. This is a two-tailed alternative hypothesis. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

8 One-Tailed Hypothesis Tests A one-tailed alternative hypothesis would be: or H 0 : β 2 = 0 H 1 : β 2 < 0 H 0 : β 2 = 0 H 1 : β 2 > 0 Gujarati and Porter list the null and alternatives as complementary (i.e., comprising all possible values) in one-tailed tests. This is not the common notation, though. Recall: Confidence intervals and hypothesis tests have a close relationship. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

9 Conducting a t-test Based on what we know, we can compute a t-distributed test statistic with n k 1 degrees of freedom: t = ˆβ 2 β 2 se( ˆβ 2 ) Our value of β 2 is drawn from the null hypothesis. Once we calculate t, we compare it to our critical value. Use the following rules based on the nature of your alternative hypothesis. H 1 : β 2 β 2 : If t > t α/2, then we reject the null hypothesis. H 1 : β 2 < β 2 : If t < t α, then we reject the null hypothesis. H 1 : β 2 > β 2 : If t > t α, then we reject the null hypothesis. Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

10 Statistical Inference: The F -Ratio ANOVA for Regression Models Besides our t-ratios, we may want to evaluate the model as a whole. Recall that the residual sum of squares is the squared set of errors: n RSS = (Y i Ŷ i ) 2, where Ŷ i is the fitted values. These are found with i=1 fitted() in R or predict in Stata. RSS has n k 1 degrees of freedom. (k =number of predictors.) Now consider the explained sum of squares, which has k degrees of freedom: ESS = n (Ŷi Ȳ )2 i=1 Define the total sum of squares as the total variation in the Y values about the mean, which has n 1 degrees of freedom: TSS = n (Y i Ȳ ) 2 i=1 Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

11 Statistical Inference: The F -Ratio Recall TSS = ESS + RSS When we divide any of these quantities by their associated degrees of freedom, it is called a mean sum of squares, and is a type of variance. Since ESS is good and RSS is bad, we can create a comparison: F = ESS/(k) RSS/(n k 1) which is distributed F with k and n k 1 degrees of freedom. Thus we have an omnibus test of whether the whole model fits well: H 0 : model does not fit at the 1-α level H 1 : model fits at the 1-α level Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

12 Predictions and Uncertainty A nice feature of regression models is they allow us to make predictions. The formula is simple: Ŷ 0 = ˆβ 1 + ˆβ 2 X 0. Beware, though, predictions that are outside of the domain of the inputs may be dicey. We would like to measure how uncertain we are about a prediction. Construct a 100(1 α) percent confidence interval: ˆβ 1 + ˆβ 2 X 0 ± t α/2 se(ŷ 0 ) Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

13 Uncertainty for Means or Individuals The key distinction: Are we expressing our uncertainty about the mean of Y given X 0? If so: [ 1 var(ŷ0) = ˆσ 2 n + (X 0 X ) 2 ] (Xi X ) 2 Alternatively, are we expressing our uncertainty about the value of a specific individual s Y given X 0? If so: [ var(ŷ 0 ) = ˆσ n + (X 0 X ) 2 ] (Xi X ) 2 Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

14 Testing for Normality in Residuals One of many assumptions that we make in the classical normal linear regression model is that the disturbances have a normal distribution. We can evaluate that validity of this assumption by examining the residuals that result from estimation. Consider three strategies: 1 Histogram of residuals 2 Normal probability plot 3 Jarque-Bera Test [ ] S 2 (K 3)2 JB = n Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15

15 For Next Time Read Gujarati & Porter chapter 6 (Extensions of the Two-Variable Linear Regression Model). Study the 2010 election data from Monogan s Dataverse. Estimate a linear model using OLS and report the results in a table as you would in a journal article. Report the 90% confidence intervals for your regression coefficients. Report the results of two hypothesis tests: 1 H 0 : β 2 = 0, H 1 : β 2 < 0 (In other words, is there a negative effect of Obama s vote share on the Republican s share?) 2 H 0 : β 2 = 1, H 1 : β 2 1 (In other words, does the average effect of Obama s vote share on the Republican s share differ from a one-to-one relationship?) Evaluate whether the residuals are normally distributed. Plot the predictions from your model with either the 90% confidence interval in predicting the mean or the interval in predicting values. (Be specific in what you re doing.) Jamie Monogan (UGA) Confidence Intervals & Hypothesis Testing POLS / 15