Indicate the life of a lamp from the new process with .Y. The mean of Y is and the standard deviation of Y is Y 200 hours. The manager accepts the alternative hypothesis as Y 2100 hours. a) The magnitude of a test is the probability of falsely rejecting a valid null hypothesis. We first calculate the probability that the manager falsely accepts the null hypothesis when it is invalid:.

## Linear Regression with One Regressor

Solutions to Odd Number End-of-Chapter Exercises 17. a) ui represents factors other than time that affect the student's performance on the exam, including amount of time to study, aptitude for the material, and so on. Some students will have studied more than average, others less; some students will have higher than average aptitude for the subject, others lower, and so on. The expectation of ˆ0 is obtained by taking expectations from both sides of Equation (4.8): YiY for all i, which implies thatˆ10, or that Xi is constant for all i.

Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

## Linear Regression with Multiple Regressors

Note that the estimator for 1 is identical to the OLS estimator from the regression of Y on X1, omitting X2. In other words, when (X1iX1)(X2iX2) 0 , the estimated coefficient on X1 in the OLS regression of Y on X1 and X2 is equal to the estimated coefficient in the OLS regression of Y on X1.

## Hypothesis Tests and Confidence Intervals in Multiple Regression

The BDR coefficient measures the partial effect of the number of bedrooms, holding the house size (Hsize) constant. If lot size were measured in thousand square feet, the rating coefficient would be 2 instead of 0.002. a) Treatment (allocation into small sections) was not randomly distributed in the population (continuing students and newly enrolled students) due to the difference in the proportion of treated continuing students and newly enrolled students. Solutions to end-of-chapter exercises 25. b) Since the treatment was randomly assigned based on enrollment status (continuing or newly enrolled), E(u | X1, X2) will not depend on X1.

## Nonlinear Regression Functions

However, since X2 was not randomly assigned (newly enrolled students may, on average, have attributes other than being newly enrolled that affect test scores), E(u | X1, X2) may depend on X2, so that ˆ2 can be biased and inconsistent. With all three binary class size variables included in the regression, it is impossible to calculate OLS estimates because the intercept is a perfect linear function of the three class size regressors. a) (1) Demand for older journals is less elastic than for younger journals because the interaction term between the log of journal age and price per citation is positive. As described in Equation (8.8) and Note on page 303, the standard error can be found by dividing 0.28, the absolute value of the estimate, by the square root of .

The signs will not change and the constant (intercept) will change. ii) The error term has a standard deviation of 2.65 (measured in logarithmic points). However, the regression does not control for many factors (firm size, industry, profitability, experience, and so on). Discrimination on the basis of gender means that two workers who are the same in all respects except gender receive different wages.

It is therefore also important to check for employee characteristics that can influence their productivity (education, number of years of experience, etc.). These are potentially important omitted variables in the regression that will lead to bias in the OLS coefficient estimator for women. Since these characteristics were not controlled for in the statistical analysis, it is premature to draw conclusions about gender discrimination. ii) Female is correlated with the two new variables included and at least one of the variables is important for explaining ln(Income).

Solutions to End-of-Chapter Exercises 27. c) Neglecting the effect or return, whose effects appear to be small and statistically insignificant, the formula for the omitted variable bias (see equation (6.1)) suggests that female is negatively related to ln(market value) .

## Assessing Studies Based on Multiple Regression

Both regressions suffer from omitted variable bias, so they will not provide reliable estimates of the causal effect of income on test scores. However, the non-linear regression in (8.18) fits the data well enough that it could be used for forecasting. Internal consistency: To the extent that price is affected by demand, there may be simultaneous equation bias.

External consistency: The Internet and the introduction of "e-journals" may cause significant changes in the academic journal market, so results for the year 2000 may not be relevant to today's market. Since all Xi are used (although some are used for incorrect values of Yj), X X and. As elsewhere in the book, we interpret n 300 as a large sample, so we use an approximation of n that tends to infinity.

Solutions for Odd Number End of Chapter Exercises 31. where the last result follows because Xi X. i for the shuffled observations and Xj is independent of Xi for i j. c) Yes, the estimator based on the first 240 observations is better than the adjusted estimator from part (b). Equation (4.21) in Key Concept 4.4 (page 171) implies that the estimator based on the first 240 observations has a variance, ie From part (a), the OLS estimator based on all the observations has two sources of sampling.

## Regression with Panel Data

To check the robustness of the linear specification, it would be useful to consider a log or a quadratic specification. Measurement error does not appear to be a problem, as variables such as traffic fatalities and taxes are accurately measured. Similarly, sample selection is not a problem because data are used from all states.

Expert knowledge is required to determine if this is a problem. a) Average snowfall does not change over time, and so will be perfectly aligned with the state fixed effect. In this case T is small (T 4), so the normal approximation by CLT is unlikely to be very good.

## Regression with a Binary Dependent Variable

Solutions to odd-numbered end-of-chapter exercises 37. b) With the P/I ratio reduced to 0.30, the probability of rejection is. For a white applicant who has a P/I ratio of 0.35, the probability that the application is the probability of being denied is 2.519. the probability of denial is 2.08 percentage points lower. In the functional form of logit regression, the marginal effect depends on the probability level which in turn depends on the race of the applicant.

The coefficient on black is 0.084, indicating an estimated rejection probability that is 8.4 percentage points higher for the black applicant. Such variables must be related to race and also be related to the probability of default on the mortgage (which in turn would lead to denial of the mortgage application). Standard measures of default probability (past credit history and employment variables) are included in the regressions shown in Table 9.2, so these omitted variables are unlikely to affect the answer in (a).

Other variables such as education, marital status, and occupation may also be related to the probability of default, and these variables are omitted from the regression in the column. Adding these variables (see columns (4)-(6)) has little effect on the estimated effect of black on the probability of mortgage rejection. one).

## Instrumental Variables Regression

The response of demand to a fall in income will be less in the short run than in the long run. There are other factors that can influence both the choice to serve in the military and the annual earnings. Another variable is "ability", which is difficult to measure, and thus difficult to control for in the regression.

The draft was determined by a national lottery, so the choice of military service was random. Because it affects the probability of serving in the military, the lottery number is important. For students in kindergarten, the estimated effect of small class treatment compared to being in a regular class is an increase of 13.90 test points with a standard error of 2.45.

For students in Grade 1, the estimated effect of small class treatment compared to being in a regular class is an increase of 29.78 test scores with a standard error of 2.83. The local network is a failure to follow the treatment protocol, and this leads to bias in the OLS estimator of the average causal effect. The treatment is to make connections available in the room; the treatment is not using the internet.

Following the notation used in Chapter 13, let 1i denote the coefficient on the state sales tax in the first stage” IV regression, and let 1i denote the cigarette demand elasticity.

## Introduction to Time Series Regression and Forecasting

Solutions for the exercises at the end of the odd numbered chapter 45. d) The conditional expectation of YT1 given YT is.

## Estimation of Dynamic Causal Effects

The dynamic causal effects are for Experiment A. The regression in Exercise 15.1 does not control for interest rates, so interest rates are assumed to evolve in their "normal pattern" given changes in oil prices.

Additional Topics in Time Series Regression

The Theory of Linear Regression with One Regressor

## ˆ 1 RLS ,

Combining the results on the numerator and denominator and applying Slutsky's theorem leads to Using the conditional mean and conditional variance of ˆ1RLS derived in parts (c) and (d), respectively, the sampling distribution of ˆ1RLS, conditional on X1,, Xn. The inequality follows by applying the Cauchy-Schwartz inequality, and the second inequality follows due to the finite fourth moments of (Xi, ui).

The finite variance together with the fact that we have the mean zero (by assumption 1 in Key Concept 15.1) and we are i.i.d. by assumption 2) implies that the sample mean v satisfies the requirements of the central limit theorem. satisfies the central limit theorem. The conditional probability distribution function of ui and Xi given uj and Xj is f (ui, Xi|uj, Xj). The second equality used the conclusion from part (a) and the independence between Xi and Xj.

In this case, the result follows directly if the nonzero exponent (r or s) is less than 4. Note: In early printing of the third edition, there was a typographical error in the expression for Y|X. The result was also shown in Exercise 13.10, and the procedure used in the exercise is discussed in part (b).

## The Theory of Multiple Regression

Solutions to the odd end-of-chapter exercises 71. where the second equality uses the fact that Q is a scalar and the third equality uses the fact that Q cw. b) Since the covariance matrix W is positive definite, we have cwc 0 for every non-zero vector from the definition.

The second equality used assumption (ii) that (,X Wi i, Y)i are i.i.d., and the third equality enforced assumption of conditional mean independence (i). e) n1X M X W converges in probability to a finite irreducible matrix, and n1X M U W converges in probability to a zero vector.