1 Independent Practice: Hypothesis tests for one parameter:

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1 1 Independent Practice: Hypothesis tests for one parameter: Data from the Indian DHS survey from 2006 includes a measure of autonomy of the women surveyed (a scale from 0-10, 10 being the most autonomous) that is based on decision-making in the household, domestic violence, the age when married, current age, a dummy for husband s education greater than primary school, and a dummy for an urban location. I estimated the following model which includes a few of these components: autonomy = β 0 + β 1 mrage + β 2 crage + β 3 husbedu + β 4 urban + u After running the above regression model we get the following results: autonomy = mrage crage husbedu urban + û (0.0400) (0.0265) (0.2835) (0.2835) (0.2735) Test the hypothesis that age at marriage has no effect on female autonomy at the 90% confidence level. Step 1: Define null and alternative: H 0 : H 1 : State clearly in your own words what the null hypothesis is claiming: When this null hypothesis holds, we know that t = which is distributed: Step 2: Write down the formula for your test statistic, plug in the values and calculate your t-stat. t = Step 3: Choose the significance level and the critical value of the test. We need three things to find c: 1. Significance level is α = 2. Two sided test or One sided test 3. Degrees of freedom: c.10 = Step 4: We reject the null hypothesis or fail to reject the null hypothesis. Step 5: Interpret in a reader friendly way: 1

2 2 Hypothesis test for multiple parameters: The F-Test Suppose we want to test that both β crage and β mrage are zero, i.e. that in general age doesn t play a significant part in autonomy. If we did these two individual t-tests, we d see that one is statistically different from zero and one isn t... so do we reject or fail to reject the null hypothesis that both are equal to zero? Don t waste your time thinking about this it s confusing, and unnecessary because the F-test does it for us Two things to remember about the F test statistic: An F-test is still a hypothesis test, so we can use all of our steps from before. The only difference is that we have a new formula for our test statistic, and we use the F-table to find the critical value instead of the t-table or normal table. To get all of the information we need for the F-test, we have to run two models: The UnRestricted Model: autonomy = β 0 + β 1 mrage + β 2 crage + β 3 husbedu + β 4 urban + u The Restricted Model: autonomy = β 0 + β 3 husbedu + β 4 urban + u (Here we restrict β 1 and β 2 to be zero) Below are the Stata results for the first UR model (again), and the results for the second R model:. reg autonomy marr_age curr_age husb_educ urban Source SS df MS Number of obs = F( 4, 971) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = autonomy Coef. Std. Err. t P> t [95% Conf. Interval] marr_age curr_age husb_educ urban _cons

3 . reg autonomy husb_educ urban Source SS df MS Number of obs = F( 2, 975) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = autonomy Coef. Std. Err. t P> t [95% Conf. Interval] husb_educ urban _cons Step 1: We need to state the null and alternative hypotheses: H 0 : H 1 : We also could have let H 1 : β 1 = 0 and/or β 2 = 0, but the first way is obviously faster. Step 2: Compute the statistic. The two formulas we have for the F-stat are: F = F = Where q is the number of restrictions: we re testing (1) β 1 = 0 and (2) β 2 = 0, so in this example q = 2. As always, n is the number of observations, which should be the same for both regressions. How do we know what k is equal to? Let s check that we know what to plug in for each formula and verify that they will equal the same thing: SSR R = SSR UR = R 2 R = R 2 UR = n = 976 k UR = q = 3

4 With the SSR definition: F = With the R 2 definition: F = ( ) / /( ) = 15.6 ( ) /2 ( ) /( ) = Step 3: Choose the significance level of the test and find the critical value from the F-table. Let s use the standard significance level in economics, 5% (Check: How often will we reject the null hypothesis even though it s true?) Notice that the only other option is 10% here, since that s all we have the tables for. To find the critical value here, we need the numerator degrees of freedom and the denominator degrees of freedom: Numerator d.o.f. q = Denominator d.o.f. n k UR 1 = c.05 = Step 4: If our F is large, larger than our critical value, this means that the RUR 2 is relatively much higher; that is, including the age variables greatly increases the amount of variation in autonomy explained by the model. If our F isn t large, it means that the R 2 for the two models are pretty close to each other, and the age variables don t have a lot of explanatory power. With this F-stat and this critical value, do we Reject the null or Fail to reject the null? Step 5: Interpret. If we reject: If we fail to reject: 4

5 3 Hypothesis Testing with Two Proportions Example Now let s consider an example from actual data for a poverty alleviation program in Mexico. In 1997, 24,059 households in rural Mexico were randomly allocated between treatment and control groups for a conditional cash transfer program called Oportunidades to keep kids in school. When analyzing the results of a randomized experiment, the first step is to verify that the control group is, on average, very much like the treatment group in terms of characteristics that we observe and have data for. For example, data was collected on household assets. Your data reveals that while 14.47% of the 14,846 treatment households have a refrigerator, and 16.53% of the 9,213 control households have one. In order to confirm that about the same proportion of households in each group have a refrigerator, we need to perform a hypothesis test. Call the sample proportion of households with a refrigerator in the treatment group ˆp t, the true treatment proportion p t, the sample proportion of households with a refrigerator in the control group ˆp c, and the true control proportion p c. Also, call the whole sample proportion of households (in either treatment or control) with a refrigerator ˆp. Step 1. H 0 : H 1 : Step 2. How do we compute this test statistic? We know that the null hypothesis specifies E[p t p c ] = 0, so what s left is the standard deviation. Whenever we re testing a difference of means, remember the formula: Var( x ȳ) = Var( x) + Var(ȳ). So applying the formula, we have that: Var( ˆp t ˆp c ) = Var( ˆp t ) + Var( ˆp c ) Var( ˆp t ) = Var( ˆp c ) = Which means SD( ˆp t ˆp c ) = The trickiest part here is keeping track of what your null hypothesis is! Now we re ready to calculate our z-statistic: ˆD = ˆp t ˆp c = ˆp = SD( ˆD) = z = 5

6 Step 3. By the null hypotheses we chose, we re doing a two-sided test. Let s choose the 5% significance level as this is the most common test that economists evaluate. Check the normal table to find that c = 1.96 Step 4. Reject Fail to reject Step 5. Interpret: At the 5% significance level, there is statistical evidence that the proportion of households with a refrigerator in the control group is not the same as the proportion of households with a refrigerator in the treatment group. What does this mean for the study? Confidence Interval With this same example, how would we compute a confidence interval? The KEY difference here is that now, instead of assuming a null hypothesis to be true, we are just taking our estimated variance from what we observe in our sample(s). Therefore, instead of constructing a ˆp that represents the mean of all observations in our sample, we allow for the means of the two samples to be different. In fact, the confidence interval is centered on our estimated difference from the sample. We then use standard errors from these sub-samples to constuct the standard error of their difference: ˆD = ˆp t ˆp c We can use the formula Var( x ȳ) = Var( x) + Var(ȳ) to find the Var( ˆD) = Var( ˆpt ˆp c ): Var( ˆD) = Var( ˆp t ) + Var( ˆp c ) Var( ˆ d p ) = s2 p n p Var( ˆ d c ) = s2 c n c You can then take the square root of this estimated variance to get an estimate for the estimator s standard error. Then, we can plug in our values in order to construct the confidence interval. CI W = [ ( ) ( )] s s x c W, x + c W n n 6