Question

Use the data in CEMENT for this exercise. (i) A static (inverse) supply function for the monthly growth in cement price $(g p r c)$ as a function of growth in quantity $(g c e m)$ is $g p r c_{t}=\alpha_{1} g c e m_{t}+\beta_{0}+\beta_{1} g p r c p e t+\beta_{2} f e b_{t}+\ldots+\beta_{12} d e c_{t}+u_{t}^{s}$ where gprcpet (growth in the price of petroleum) is assumed to be exogenous and $f e b, \ldots,$ dec are monthly dummy variables. What signs do you expect for $\alpha_{1}$ and $\beta_{1} ?$ Estimate the equation by OLS. Does the supply function slope upward? (ii) The variable $g$ defs is the monthly growth in real defense spending in the United States. What do you need to assume about gdefs for it to be a good IV for gcem? Test whether gcem is partially correlated with gdefs. (Do not worry about possible serial correlation in the reduced form.) Can you use $g d e f s$ as an IV in estimating the supply function? (iii) Shea $(1993)$ argues that the growth in output of residential (gres) and nonresidential (gnon) construction are valid instruments for gcem. The idea is that these are demand shifters that should be roughly uncorrelated with the supply error $u_{t}^{s} .$ Test whether gcem is partially correlated with gres and gnon; again, do not worry about serial correlation in the reduced form. (iv) Estimate the supply function, using gres and gnon as IVs for gcem. What do you conclude about the static supply function for cement? [The dynamic supply function is, apparently, upward sloping; see Shea $(1993) . ]$

Name: Problem 6, Chapter 16: Introductory Econometrics
Uploaded: 2021-03-17T01:45:24-08:00
Duration: 6 min 46 s
Channel: David Gagnon

   Use the data in CEMENT for this exercise.
(i) A static (inverse) supply function for the monthly growth in cement price $(g p r c)$ as a function of growth in quantity $(g c e m)$ is
$g p r c_{t}=\alpha_{1} g c e m_{t}+\beta_{0}+\beta_{1} g p r c p e t+\beta_{2} f e b_{t}+\ldots+\beta_{12} d e c_{t}+u_{t}^{s}$
where gprcpet (growth in the price of petroleum) is assumed to be exogenous and $f e b, \ldots,$ dec are monthly dummy variables. What signs do you expect for $\alpha_{1}$ and $\beta_{1} ?$ Estimate the equation by OLS. Does the supply function slope upward?
(ii) The variable $g$ defs is the monthly growth in real defense spending in the United States. What do you need to assume about gdefs for it to be a good IV for gcem? Test whether gcem is partially correlated with gdefs. (Do not worry about possible serial correlation in the reduced form.) Can you use $g d e f s$ as an IV in estimating the supply function?
(iii) Shea $(1993)$ argues that the growth in output of residential (gres) and nonresidential (gnon) construction are valid instruments for gcem. The idea is that these are demand shifters that should be roughly uncorrelated with the supply error $u_{t}^{s} .$ Test whether gcem is partially correlated with gres and gnon; again, do not worry about serial correlation in the reduced form.
(iv) Estimate the supply function, using gres and gnon as IVs for gcem. What do you conclude about the static supply function for cement? [The dynamic supply function is, apparently, upward sloping; see Shea $(1993) . ]$

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 16, Problem 6

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Solved by Expert David Gagnon

03/17/2021

Step 1

Then, we create the supply curve using the given equation. The equation is estimated using Ordinary Least Squares (OLS) method. The estimated coefficients are $\alpha_{1} = -0.044254$ and $\beta_{1} = -1.054093$. Show more…

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Use the data in CEMENT for this exercise. (i) A static (inverse) supply function for the monthly growth in cement price $(g p r c)$ as a function of growth in quantity $(g c e m)$ is $g p r c_{t}=\alpha_{1} g c e m_{t}+\beta_{0}+\beta_{1} g p r c p e t+\beta_{2} f e b_{t}+\ldots+\beta_{12} d e c_{t}+u_{t}^{s}$ where gprcpet (growth in the price of petroleum) is assumed to be exogenous and $f e b, \ldots,$ dec are monthly dummy variables. What signs do you expect for $\alpha_{1}$ and $\beta_{1} ?$ Estimate the equation by OLS. Does the supply function slope upward? (ii) The variable $g$ defs is the monthly growth in real defense spending in the United States. What do you need to assume about gdefs for it to be a good IV for gcem? Test whether gcem is partially correlated with gdefs. (Do not worry about possible serial correlation in the reduced form.) Can you use $g d e f s$ as an IV in estimating the supply function? (iii) Shea $(1993)$ argues that the growth in output of residential (gres) and nonresidential (gnon) construction are valid instruments for gcem. The idea is that these are demand shifters that should be roughly uncorrelated with the supply error $u_{t}^{s} .$ Test whether gcem is partially correlated with gres and gnon; again, do not worry about serial correlation in the reduced form. (iv) Estimate the supply function, using gres and gnon as IVs for gcem. What do you conclude about the static supply function for cement? [The dynamic supply function is, apparently, upward sloping; see Shea $(1993) . ]$

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Key Concepts

Inverse Supply Function

An inverse supply function is a representation where price is expressed as a function of quantity rather than the traditional quantity as a function of price. This concept is important as it can help determine whether the supply condition holds, such as an upward?sloping relationship, meaning that as the quantity supplied increases, the price also increases. The signs of key coefficients in this context can indicate whether the supply function behaves in accordance with economic theory.

Ordinary Least Squares (OLS) Estimation

OLS estimation is a fundamental technique in regression analysis used to estimate the relationships between variables by minimizing the sum of squared residuals. In the context of supply functions, OLS provides baseline estimates of the relationship between price and quantity, though caution is needed if there is concern about endogeneity or omitted variable bias.

Instrumental Variables (IV) Estimation

IV estimation is used when one or more regressors are potentially endogenous, meaning they are correlated with the error term, which can bias OLS estimates. IV methods rely on finding instruments—variables that are correlated with the endogenous regressor but uncorrelated with the error term in the main equation—to provide consistent parameter estimates.

Validity Conditions for Instruments

For an instrument to be valid, it must satisfy two core conditions: relevance and exogeneity. Relevance means the instrument must be sufficiently correlated with the endogenous explanatory variable, while exogeneity requires that the instrument is not correlated with the error term in the primary regression model. These conditions ensure that the instrument can properly isolate the variation in the endogenous regressor that is not contaminated by the error term.

Testing Instrument Relevance

Testing for instrument relevance involves checking if the instrument is partially correlated with the endogenous variable. This is typically performed by estimating a first-stage regression and testing whether the coefficient on the instrument is statistically different from zero. A significant partial correlation indicates that the instrument has explanatory power for the endogenous variable, which is essential for the success of the IV estimation.

Seasonal/Monthly Dummy Variables

Seasonal or monthly dummy variables are used in regression analysis to control for fixed effects that recur over specific time intervals, such as months. Inclusion of these dummies helps to absorb seasonality in the data, ensuring that the estimation of primary coefficients is not biased by systematic variations that occur at regular intervals over time.

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Transcript

00:03 All right, hey guys, welcome to another econometrics tutorial, and in this one, we're going to be making a supply function and using some instrumental variables to help predict the quantity a bit more accurately.

00:22 So, sorry, the slope of the supply curve a bit more accurately.

00:24 So, of course, you're going to want to start by opening up the package, woodridge, and then the dataset cement, and then you're going to want to want to create.

00:35 Create your supply curve.

00:37 So, okay, first a little bit about the data for cement.

00:41 So the package here has a lot of, or the data set has a lot of data on cement, as you might imagine, by the name.

00:51 The important ones we're going to be using are the growth in the price of cement, which is gprc, the growth in the quantity of cement, which is gcem, and then gpr, c -p -e -t is the growth in the price of crude petroleum.

01:14 And then here we have a bunch of dummy variables for each month because the growth rates are calculated on a monthly basis, so it's good to have that along the side.

01:26 So let's go ahead and run the linear model, and let's see what rb2 is, which would be the coefficient for our...

01:38 Growth in cement quantity, which for the supply curve is the slope.

01:45 So over here, our b2 is, so your basic economic knowledge might be screaming out at you, because typically when you think of supply curves, they're upward sloping.

01:58 But here, our b2, so our slope of our supply curve is actually negative.

02:03 It's negative 0 .04.

02:05 4.

02:06 So that is indeed quite weird.

02:10 So to see if we can maybe find out if there's anything else affecting the quantity here, we're going to use some instrumental variables.

02:21 So first, of course, you have to open up the aer package to be able to use the instrumental variable function in r.

02:29 So do that.

02:30 And then we'll make our instrumental variable equation here.

02:35 So if you might remember from some of past videos you write your equation as normal and then you put a vertical line there and then you write the reduced form so what that would mean is basically write the equation the same again but remove what you want to be using instruments for and then add the instruments on the end so in this case we're going to be using g -defs which is the growth in monthly government spending on defense and then we'll we'll get the summary of that and the diagnostics to have a look at the instrumental variable tests.

03:17 All right.

03:17 So since we just have one, we don't have to worry about the sargon test.

03:22 Well, right off the bat, the weak instruments test.

03:25 We're not rejecting the null hypothesis.

03:27 So government defense is a weak instrument for the growth rate in cement quantity, which kind of makes sense...

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