Question

Use SMOKE for this exercise. (i) A model to estimate the effects of smoking on annual income (perhaps through lost work days due to illness, or productivity effects) is $\log ($ income $)=\beta_{0}+\beta_{1}$ cigs $+\beta_{2} e d u c+\beta_{3} a g e+\beta_{4} a g e^{2}+u_{1}$ where cigs is number of cigarettes smoked per day, on average. How do you interpret $\beta_{1} ?$ (ii) $\quad$ To reflect the fact that cigarette consumption might be jointly determined with income, a demand for cigarettes equation is $\begin{aligned} c i g s=& \gamma_{0}+\gamma_{1} \log (\text {income})+\gamma_{2} e d u c+\gamma_{3} a g e+\gamma_{4} a g e^{2} \\ &+\gamma_{5} \log (\text {cigpric})+\gamma_{6} r e s t a u r n+u_{2} \end{aligned}$ where cigpric is the price of a pack of cigarettes (in cents) and restaurn is a binary variable equal to unity if the person lives in a state with restaurant smoking restrictions. Assuming these are exogenous to the individual, what signs would you expect for $\gamma_{5}$ and $\gamma_{6} ?$ (iii) Under what assumption is the income equation from part (i) identified? (iv) Estimate the income equation by OLS and discuss the estimate of $\beta_{1 .}$ (v) Estimate the reduced form for cigs. Recall that this entails regressing cigs on all exogenous variables.) Are log(cigpric) and restaurn significant in the reduced form? (vi) Now, estimate the income equation by 2 $\mathrm{SLS}$ . Discuss how the estimate of $\beta_{1}$ compares with the OLS estimate. (vii) Do you think that cigarette prices and restaurant smoking restrictions are exogenous in the income equation?

Name: Problem 1, Chapter 16: Introductory Econometrics
Uploaded: 2021-02-22T16:38:27-08:00
Duration: 7 min 19 s
Channel: Kaylee Mcclellan

   Use SMOKE for this exercise.
(i) A model to estimate the effects of smoking on annual income (perhaps through lost work days due to illness, or productivity effects) is
$\log ($ income $)=\beta_{0}+\beta_{1}$ cigs $+\beta_{2} e d u c+\beta_{3} a g e+\beta_{4} a g e^{2}+u_{1}$
where cigs is number of cigarettes smoked per day, on average. How do you interpret $\beta_{1} ?$
(ii) $\quad$ To reflect the fact that cigarette consumption might be jointly determined with income, a demand for cigarettes equation is
$\begin{aligned} c i g s=& \gamma_{0}+\gamma_{1} \log (\text {income})+\gamma_{2} e d u c+\gamma_{3} a g e+\gamma_{4} a g e^{2} \\ &+\gamma_{5} \log (\text {cigpric})+\gamma_{6} r e s t a u r n+u_{2} \end{aligned}$
where cigpric is the price of a pack of cigarettes (in cents) and restaurn is a binary variable equal to unity if the person lives in a state with restaurant smoking restrictions. Assuming these are exogenous to the individual, what signs would you expect for $\gamma_{5}$ and $\gamma_{6} ?$
(iii) Under what assumption is the income equation from part (i) identified?
(iv) Estimate the income equation by OLS and discuss the estimate of $\beta_{1 .}$
(v) Estimate the reduced form for cigs. Recall that this entails regressing cigs on all exogenous variables.) Are log(cigpric) and restaurn significant in the reduced form?
(vi) Now, estimate the income equation by 2 $\mathrm{SLS}$ . Discuss how the estimate of $\beta_{1}$ compares with the OLS estimate.
(vii) Do you think that cigarette prices and restaurant smoking restrictions are exogenous in the income equation?

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 16, Problem 1

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Solved by Expert Kaylee Mcclellan

02/22/2021

Step 1

This coefficient represents the change in the log of income for each additional cigarette smoked per day. Because the dependent variable is in logarithmic form, the coefficient represents a percentage change. Therefore, an increase in the number of cigarettes Show more…

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Use SMOKE for this exercise. (i) A model to estimate the effects of smoking on annual income (perhaps through lost work days due to illness, or productivity effects) is $\log ($ income $)=\beta_{0}+\beta_{1}$ cigs $+\beta_{2} e d u c+\beta_{3} a g e+\beta_{4} a g e^{2}+u_{1}$ where cigs is number of cigarettes smoked per day, on average. How do you interpret $\beta_{1} ?$ (ii) $\quad$ To reflect the fact that cigarette consumption might be jointly determined with income, a demand for cigarettes equation is $\begin{aligned} c i g s=& \gamma_{0}+\gamma_{1} \log (\text {income})+\gamma_{2} e d u c+\gamma_{3} a g e+\gamma_{4} a g e^{2} \\ &+\gamma_{5} \log (\text {cigpric})+\gamma_{6} r e s t a u r n+u_{2} \end{aligned}$ where cigpric is the price of a pack of cigarettes (in cents) and restaurn is a binary variable equal to unity if the person lives in a state with restaurant smoking restrictions. Assuming these are exogenous to the individual, what signs would you expect for $\gamma_{5}$ and $\gamma_{6} ?$ (iii) Under what assumption is the income equation from part (i) identified? (iv) Estimate the income equation by OLS and discuss the estimate of $\beta_{1 .}$ (v) Estimate the reduced form for cigs. Recall that this entails regressing cigs on all exogenous variables.) Are log(cigpric) and restaurn significant in the reduced form? (vi) Now, estimate the income equation by 2 $\mathrm{SLS}$ . Discuss how the estimate of $\beta_{1}$ compares with the OLS estimate. (vii) Do you think that cigarette prices and restaurant smoking restrictions are exogenous in the income equation?

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Key Concepts

Endogeneity in Econometric Models

Endogeneity occurs when an explanatory variable is correlated with the error term, often due to omitted variables, measurement error, or simultaneous causality. This issue can bias the estimates obtained via ordinary least squares (OLS), leading to incorrect inferences about causal relationships.

Instrumental Variables (IV)

Instrumental variables are used to address endogeneity by providing a source of variation in the endogenous explanatory variable that is uncorrelated with the error term. A valid instrument must be both relevant, meaning it is correlated with the endogenous regressor, and exogenous, meaning it is not correlated with the regression error term.

Reduced Form Equations

Reduced form equations express the endogenous variables solely as functions of exogenous variables and the error term. They are used to study the total effect of exogenous shocks on the system and are an essential part of instrumental variable analysis, helping to verify the relevance and exogeneity of instruments.

Identification in Simultaneous Equations Models

Identification in simultaneous equations models refers to the conditions that allow for unique estimation of the model parameters. Sufficient variation and proper exclusion restrictions, where some exogenous variables affect one equation but not another directly, are key for achieving identification.

Two-Stage Least Squares (2SLS)

Two-Stage Least Squares is an estimation technique used in the presence of endogeneity. In the first stage, the endogenous variable is regressed on all exogenous variables, including the instruments, to obtain predicted values. In the second stage, the predicted values replace the endogenous variable in the original regression, yielding consistent estimates of the parameters.

Exogeneity Assumptions

Exogeneity assumptions are essential in econometric modeling, particularly with instrumental variables. Exogenous instruments must not be correlated with the error term in the structural equation; violating this assumption can lead to inconsistent and biased estimates, undermining the causal interpretation of the model parameters.

Interpretation of Coefficients in Log-Linear Models

In log-linear regression models, the coefficients are interpreted as approximate percentage changes in the dependent variable for a one-unit change in the independent variable, holding other factors constant. This interpretation is critical when the dependent variable has been logarithmically transformed, as it captures multiplicative rather than additive effects.

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Transcript

00:01 Okay, so today we are going to be working with simultaneous equations models, starting really simple with the two equations you can see here, equation one and equation two.

00:12 Equation one is basically just estimating the effect of cigarette consumption on annual income, whereas equation two is assuming that annual income and cigarette consumption, those two variables, may be jointly determined, in which case our second equation is estimating the demand for cigarette.

00:31 As a function of annual income along with a variety of other variables that we've included.

00:38 To begin, let's start by interpreting beta 1 in our first equation, just to understand exactly what that means there.

00:47 So for beta 1, because you can see that this is in a semi -logorhythmic form, our dependent variable is log of income.

00:54 We can assume that per each unit change in beta 1, we're going to see that percent change.

01:01 In our dependent variable.

01:04 So to determine that change exactly, we would take beta 1 times 100 to get whatever percentage change in income.

01:16 That would be percentage change in income.

01:25 All right.

01:26 And then let's go ahead and take a look now at our second equation and determine what our expected signs might be of the variables that we've added here.

01:38 So our gamma 5 and gamma 6, gamma 5 depicting our percent change in price of cigarettes and gamma 6 being a binary variable or a dummy variable indicating whether or not a location has restrictions on smoking within restaurants.

02:00 So determining each of these, we can, based upon economic theory, we know that as price goes up, demand tends to go down for that particular product, in which case we can infer from that, that gamma -5 has an expected sign or an a -priorri -expected sign, which is negative.

02:23 It should have a negative impact on cigarette consumption.

02:27 And then for our gamma -6 here, for whether or not there are smoking restrictions within a restaurant, we can also assume that that has an a priori negative expected sign because if people are unable to smoke in a restaurant, it's likely that they'll consume fewer cigarettes because there's a reduced number of places in which they can actually smoke those cigarettes.

02:51 From this, we're able to infer that we're working under the assumption that both gamma, that either gamma 5 is not equal to zero, or that gamma 6 is not equal to 0.

03:09 Because if they were equal to 0, there would be no point including them in our equation, in which case they would have absolutely no effect or no significant effect on our dependent variable.

03:20 If we were to estimate our first equation using ordinary least squares method, what we would end up finding is that the actual value of beta 1 would be equal to 0 .00 .0 .000.

03:37 1731...

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