How to proceed when both the dependent and the explanatory variables should be subjected to a first-stage regression?

[Fernando Carnot]

Dear forum members,

On "Consumption, Income, and Interest Rates: Reinterpreting the Time Series Evidence", Campbell and Mankiw use instrumental variables to regress Δc(t) on Δy(t) [Δc(t) = μ + λΔy(t)], where Δc(t) is the difference between the log of the consumption in t and the log of the consumption in t-1, and Δy(t) is the difference between the log of the income in t and the log of the income in t-1.

Nonetheless, both Δc(t) and Δy(t) are subjected to a first-stage regression on the same set of instrumental variables. I mean:

Δc(t) = β0 + β1X1(1t) + ... + βkX(kt) + η(ct)

Δy(t) = γ0 + γ1X1(1t) + ... + γkX(kt) + η(yt)

My question is: How do I perform this two stage least squares regression once both the dependent and the explanatory variables are subjected to a first-stage regression?

I know how to proceed on a "regular" two-stage least squares regression, when one or more explanatory variables are subjected to the first-stage step, but regressing the dependent variable is something I've never seen.

How should I proceed on Stata to perform this kind of regression?

Thank you very much.

[Jeff Wooldridge]

The two estimates are identical, and so you should just use the usual 2SLS apparatus.

[Fernando Carnot]

Thank you for your answer, Jeff.

I would like to know how to perform this kind of regression on Stata.

I know that if I want to regress the dependent variable y on the explanatory variable x using z as an instrumental variable I can use the ivregress command the following way:

ivregress 2sls y (x = z)

However, I don't know how to proceed when I have to use an instrumental variable for y.

According to Stata manual the ivregress syntax is:

ivregress estimator depvar [ varlist1] (varlist2 = varlistiv) [ if ] [ in ] [weight ] [ , options]

So (as far as I understand) I cannot perform this kind of 2sls through this command. What is the best option?

Thanks again.

[Andrew Musau]

The referenced article attempts to test the permanent income hypothesis: https://scholar.harvard.edu/files/ma...ion_income.pdf. Read pages 188-189. Equation 1.4 is what needs to be estimated using 2SLS. What you show are separately estimated using linear regression and the same hypothesis is tested from the derived estimates, where the rationale for this is explained.

[Fernando Carnot]

Thank you, Andrew.

I thought I understood but it seems I did not.

Both Δc and Δy were subjected to OLS regressions (first stage) on the instruments (as can be seen on page 192). Then (if I understood your post) the regressions' predicted values were used to estimate λ on a third OLS regression.

I mean:

regress delta_c IVs
predict delta_c_predicted, xb

regress delta_y IVs
predict delta_y_predicted, xb

regress delta_c_predicted delta_y_predicted

Is this correct?

[Andrew Musau]

Look at the foot of the results table in page 192.

It explicitly tells you that "the column labeled "\(\lambda\) estimate" reports the IV estimate of \(\lambda\). Again, the authors state that they estimated Eq. 1.4 using IV regression. Here is equation 1.4

Now, write the equation as

$$\Delta C_t= \lambda \Delta Y_t + u_t$$

where \(u_t = (1-\lambda)\epsilon_t\). It is apparent that \(\lambda\) in the above equation is simply the parameter on \(\Delta Y_t\). So estimate the above equation using IV regression and obtain the coefficient on \(\Delta Y_t\). Naturally, you have data on income and consumption over time, so generating the differenced variables is not an issue. The authors specify exactly what instruments they use, so read the article and generate these instruments. I have a lot of teaching this semester, so I will not get deeper into this.

[Fernando Carnot]

Dear Andrew,

I'm a newbie working under a very tight deadline.

I appreciate your answer and the the time you spent helping me on this matter.

Have a great teaching semester.