Эконометрика — Совбак ВШЭ и РЭШ, 2020 final

Harvard economist Claudia Goldin attributes much of the rise of professional women in the U.S. labor force to their ability to engage in family planning after the introduction of the birth-control pill. In developing countries, early childbearing is associated with lower education and greater dependence on husbands' earnings.

This part studies the effect of family size on female labor supply using $n=254{,}654$ married women aged 21-35 from the 1980 U.S. Census. The data refer to calendar year 1979.

Table 1. Variables

Table 2. Child sex composition, family size, and labor supply

Robust standard errors are in parentheses. All regressions include an intercept, not reported. $^{**}$ denotes 1% significance and $^*$ denotes 5% significance.

1 — 3 points

Give the best reason why the OLS estimator of the coefficient on Kids $>2$ in column (3) may be biased.

2 — 3 points

Consider the hypothesis that, on average, U.S. parents want children of both sexes. Does Table 2 provide evidence for this hypothesis, against it, or neither? Explain.

3 — 6 points

Consider each proposed instrument for Kids $>2$ in regression (3). Is it arguably valid? Explain.

1. (3 points) Whether the wife came from a large family.
2. (3 points) The teenage-pregnancy rate in the wife's city or town.

4 — 6 points

Using judgment and the empirical results in Table 2:

1. (3 points) Is Same sex a valid instrument in regression (4)?
2. (3 points) Are 2 boys and 2 girls a valid instrument set in regression (5)?

5 — 3 points

The estimated coefficient on Kids $>2$ is more negative in OLS regression (3) than in TSLS regression (4). Give a real-world interpretation that could explain this difference.

Consider the hypothetical regression

$$WifeWeeks_i=\beta_0+\beta_1(Kids>2)_i+u_i, \tag{7}$$

estimated by TSLS using Same sex as the instrument. For this question, assume Same sex is valid in regression (4) and is independent of every control in regression (4), so that, for example,

$$E(BoyFirst\mid SameSex)=E(BoyFirst),$$

and analogously for the other controls.

1 — 7 points

1. (3.5 points) Explain why Same sex is a valid instrument in regression (7).
2. (3.5 points) Despite its validity in regression (7), why might regression (4) still be preferable?

2 — 4 points

Suppose the labor-supply effect of having a large family differs across women: the more professionally ambitious a woman is, the smaller the effect, and the most ambitious women work whether or not they have a large family. How does this affect interpretation of regressions (4) and (5)?

Use Table 2 to assess each statement.

3 — 4 points

Families with many children may be unusual because of religious or ethnic background. Therefore, regressions (4) and (5) do not estimate the effect of family size on labor supply; they only capture religious or ethnic effects. Agree or disagree, with a specific explanation.

4 — 4 points

Even if large families reduce female labor-force participation, husbands work more to compensate for the loss of the wife's earnings. Agree or disagree using Table 2.

Do smoking bans in bars reduce smoking? The data are a panel of 50 U.S. states observed from 2001 through 2009, for $50\times9=450$ state-year observations.

Table 3. Variable definitions and summary statistics

Table 4. Smoking rates and public smoking bans

Dependent variable: smokingrate. Standard errors are in parentheses. Regressions (1)-(2) use 2009 only; regressions (3)-(6) use all years. Regressions (3)-(6) include state fixed effects. Regressions (4)-(6) also include year fixed effects. Standard errors are heteroskedasticity-robust in (1)-(2) and clustered by state in (3)-(6).

1

Using regression (2):

1. (2 points) Interpret the coefficient on statebarban.
2. (2 points) Construct a 95% confidence interval for the population coefficient.

2 — 2.5 points

Give a reason why the statebarban coefficient changes between regressions (1) and (2), including the direction of the change.

3 — 2.5 points

Give a reason why the statebarban coefficient changes between regressions (3) and (4), including the direction of the change.

4

Using regression (4):

1. (2 points) Test at the 5% level whether all coefficients on educational-achievement variables are zero.
2. (2 points) Are the estimated education-related differences in smoking rates large or small in a real-world sense?

5 — 2 points

Regression (5) includes all3bans, which equals the product of statebarban, staterestban, and stateworkban. Does regression (5) suffer from perfect multicollinearity? Explain.

6

Using regression (6):

1. (2 points) Compute the predicted effect of a bar-smoking ban when drinkingrate is 0.70.
2. (2 points) Explain precisely how to construct a 95% confidence interval for that predicted effect. A numerical interval is not required.

1 — 5 points

Consider

$$Y_{it}=\beta_1X_{it}+\alpha_i+\lambda_it+u_{it},$$

where $t=1,\ldots,T$, $i=1,\ldots,n$, and $\alpha_i+\lambda_it$ is an unobserved entity-specific time trend. How would you estimate $\beta_1$?

2. Linear probability model — 8 points

Consider

$$Y_i=\beta_0+\beta_1X_i+u_i, \qquad E(u_i\mid X_i)=0.$$

1. (1 point) Show that $P(Y_i=1\mid X_i)=\beta_0+\beta_1X_i$.
2. (2 points) Show that $$\operatorname{Var}(u_i\mid X_i) =(\beta_0+\beta_1X_i)\bigl[1-(\beta_0+\beta_1X_i)\bigr].$$
3. (1 point) Is $u_i$ heteroskedastic? Explain.
4. (4 points) Derive the likelihood function.

3. Two instruments — 5 points

A model has one endogenous regressor $X_i$ and two instruments $Z_{1i}$ and $Z_{2i}$. There is a strong theoretical reason for $Z_{1i}$ to be exogenous because it results from a random lottery, but $Z_{1i}$ alone is weak. Instrument $Z_{2i}$ is strongly relevant but less likely to be exogenous. With both instruments, the overidentification statistic is

$$J=7.5.$$

1. (2.5 points) Does this suggest $E(u_i\mid Z_{1i},Z_{2i})\neq0$? Explain.
2. (2.5 points) Does it suggest $E(u_i\mid Z_{2i})\neq0$? Explain.

4. Omitted controls in IV — 5 points

One student estimates

$$Y_i=\beta_0+\beta_1X_i+\beta_2W_i+u_i$$

using $Z_i$ as an instrument. Another estimates the same relationship but omits $W_i$.

1. (2.5 points) The first student says that if $Z_i$ and $W_i$ are correlated, the second student's IV estimator is inconsistent. Is this correct?
2. (2.5 points) The second student says that if the true $\beta_2=0$, the IV estimator remains consistent. Is this correct?

5. Forecasting an AR(1) — 5 points

Consider the stationary model

$$Y_t=\beta_0+\beta_1Y_{t-1}+u_t,$$

where $u_t$ is i.i.d. with mean zero and variance $\sigma_u^2$. Using observations $t=1,\ldots,T$, the forecast is

$$\widehat Y_{T+1\mid T}=\widehat\beta_0+\widehat\beta_1Y_T.$$

1. (1 point) Show that $$Y_{T+1}-\widehat Y_{T+1\mid T} =u_{T+1}-\bigl[(\widehat\beta_0-\beta_0)+(\widehat\beta_1-\beta_1)Y_T\bigr].$$
2. (1 point) Show that $u_{T+1}$ is independent of $Y_T$.
3. (1 point) Show that $u_{T+1}$ is independent of $\widehat\beta_0$ and $\widehat\beta_1$.
4. (2 points) Show that $$\operatorname{Var}(Y_{T+1}-\widehat Y_{T+1\mid T}) =\sigma_u^2+\operatorname{Var}\bigl[(\widehat\beta_0-\beta_0)+(\widehat\beta_1-\beta_1)Y_T\bigr].$$

6. Random walk — 6 points

Suppose

$$Y_t=Y_{t-1}+u_t, \qquad Y_0=0,$$

where $u_t$ is i.i.d. with mean zero and variance $\sigma_u^2$.

1. (2 points) Compute the mean and variance of $Y_t$.
2. (2 points) Compute $\operatorname{Cov}(Y_t,Y_{t-k})$.
3. (2 points) Use the results to show that $Y_t$ is nonstationary.

7. OLS with serially correlated regressors and errors — 7 points

Consider

$$Y_t=\beta_0+\beta_1X_t+u_t,$$

where

$$u_t=\phi_1u_{t-1}+\widetilde u_t, \qquad |\phi_1|<1,$$

and

$$X_t=\gamma_1X_{t-1}+e_t, \qquad |\gamma_1|<1.$$

The innovations $\widetilde u_t$ and $e_t$ are i.i.d. with variances $\sigma_{\widetilde u}^2$ and $\sigma_e^2$, and $e_t$ is independent of $\widetilde u_{t+i}$ for all $t$ and $i$.

1. (1 point) Show that $$\operatorname{Var}(u_t)=\frac{\sigma_{\widetilde u}^2}{1-\phi_1^2}, \qquad \operatorname{Var}(X_t)=\frac{\sigma_e^2}{1-\gamma_1^2}.$$
2. (1 point) Show that $$\operatorname{Cov}(u_t,u_{t-j})=\phi_1^j\operatorname{Var}(u_t), \qquad \operatorname{Cov}(X_t,X_{t-j})=\gamma_1^j\operatorname{Var}(X_t).$$
3. (1 point) Show that the corresponding correlations are $\phi_1^j$ and $\gamma_1^j$.
4. (4 points) Find the asymptotic variance of $\widehat\beta^{OLS}_1$.