Homework (Mathematical Econometrics)
1. [32 points] You observe an iid sample of data (Yi, Li, Ki) across a set of manufacturing firms i. Here Yi denotes the output (e.g. total sales) of the firm in some period, Li measures the labor input (e.g. total wage bill) of the firm in this period, and Ki measures the capital input (e.g. total value of machines and other assets) of the firm in this period. We are interested in estimating a production function: i.e. the structural relationship determining a firm’s ability to produce output given a set of inputs.
(a) [6 points] Suppose you estimate a regression of In Yi on In Li and In Ki (and a constant), where In denotes the natural log. Explain how you would interpret the estimated coefficients on In Li and In Ki, without making any assumptions on the structural relationship.
(b) [8 points] Now suppose you assume a Cobb-Douglas production function: Yi = Qi1,71q for some parameters (a, 13), where Qi denotes the (unobserved) productivity of firm i. Suppose we assume productivity shocks are as-good-as-random across firms: i.e. that Qi is independent of (Li, Ki). Show that under this assumption the regression estimated in (a) identifies a and /3.
(c) [8 points] Suppose we further assume constant returns-to-scale: a+/3 = 1. Show that a bivariate regression of ln(Yi/Li) on ln(Ki/Li) (and a constant) identifies the production function parameters, maintaining the independence assumption in (b). How could we test the constant-returns-to-scale assumption here?
(d) [10 points] Let’s now weaken the as-good-as-random assigment assumption in (b). Suppose we model Qi = Sfei where Si denotes the observed size of firm i, 0 is a parameter governing the relationship between firm size and productivity, and Ei is a productivity shock that is independent of (Si, Li, Ki). Specify a regression which identifies and 0 under this assumption, maintaining the assumption of a + /3 = 1. Do you expect the regression estimated in (c) to overstate or understate /3, given the new model?
2. [32 points] Suppose we are interested in estimating the (potentially different) employment effects of minimum wage increases for high school dropouts and high school graduates. As in Card and Krueger (1994), we observe employment outcomes for a sample of individuals of both educational groups in New Jersey and Pennslyvariia, before and after the New Jersey minimum wage increase. Let Yit denote the employment status of individual i at time t, let Di E {0, 1} indicate an individual’s residence in New Jersey (asumirig nobody moves between the two time periods), and let Postt E {O, 1} indicate the latter time period. Furthermore let Gradi E 10,11 indicate high school graduation. Consider the regression of
Yit =/-L + aDi + rPostt + 7Gradi + ADiPostt (1) + APosttGradi + IPDiGradi + 7DiGradiPostt + vit•
Note in that this regression includes all “main effects” (Di, Postt, and Gradi), all two-way interactions (DiPostt, PosttGradi, and DiGradi) as well as the three-way interaction DiGradiPostt.
(a) [7 Points] Suppose we regress Yit on Di, Postt, and DiPostt in the sub-sample of high school dropouts (with Gradi = 0). Derive the coefficients for this sub-sample regression in terms of the coefficients in the full-sample regression (1). Repeat this exercise for the saturated regression of Yit on Di, Postt, and DiPostt in the sub-sample of high school graduates (with Gradi = 1): what do the coefficients for this sub-sample regression equal, in terms of the coefficients in (1)?
(b) [8 Points] Extending what we saw in lecture, state assumptions under which these two sub-sample regressions (in the Gradi = 0 and Grade = 1 subsamples) identify the causal effects of minimum wage increases on employment for high school dropouts and graduates, respectively. Prove your claims.
(c) [7 Points] Under the assumptions in (b), which coefficient in (1) yields a test for whether the minimum wage effects for high school dropouts and graduates differ? Use your answers in (a).
(d) [10 Points] Suppose New Jersey and Pennslyvariia were on different employment trends when the minimum wage was increased, such that your assumptions in (b) fail. However, suppose the difference in employment trends across states is the same for high school dropouts and graduates. Show that under this weaker assumption the coefficient from (c) still identifies the difference in minimum wage effects across the groups.
