CHAPTER 6

Panel Data Techniques

Taila asks, “Prof. Empirie introduced us to longitudinal/panel datasets in chapter 2, but we have not discussed anything on using them. I was wondering if you could provide us with more information.” Prof. Metric responds enthusiastically that we will learn panel data analysis this week. He says that upon finishing this chapter, we will be able to:

Nature of Panel Data

Prof. Metric reminds the class that a panel dataset combines a cross-sectional dataset with a time-series dataset. Invo asks, “Why does one have to learn panel data techniques?” Taila volunteers to give an example of the advantage of using panel data. If we have a time-series dataset on pajama sales in Korea for 15 months and another dataset on pajama sales in China for the same 15 months, then combining the two datasets gives us a sample size of 30 data points.

Prof. Metric praises Taila on offering a good example and says that another advantage of using the panel data technique is that we will be able to observe more than one identity over time to control for the individual heterogeneity. In this particular case, the observation provides us with additional information on the specific characteristics of each market. For example, we can understand demand for pajamas in Northeast Asia by studying pajama sales in Korea and China. In addition, we are able to carry out a comparative study over a period of time. For example, we can compare demand in Korea over the past five years with demand in China during the same time period and then develop different strategies to increase sales in each country.

Panel data can be divided into “short-and-wide” panels or “long-and-narrow” panels. If I is the number of individuals observed in each of T time periods, then a short-and-wide panel has I > T while a long-and-narrow panel has T > I. Panel data can also be divided into balanced and unbalanced panels, with the unbalanced panels having some observations missing, while the balanced ones do not have any missing observation. Prof. Metric says that Excel cannot handle missing observations, so we need to delete the whole row if we run into this problem.

We learn that we can use OLS to run regression on a panel dataset if the two identities have the same characteristics; for example, if pajama sales in Korea and China have the same pattern, then the coefficient estimates will be the same for the two countries. If this is the case, all we have to do is stack one dataset above the other and then run a regression called a “pooled OLS” estimation to extend the dataset to 30 data points, as mentioned by Taila, so that CLT will guarantee valid test results. Hence, the model for the pooled OLS is:

Prof. Metric reminds us to notice that the parameters a₁, a₂, and a₃, are still the same as those in Chapter 3, even though each of the variables and the error term have the subscript “it” added.

a_1it = a₁; a_2it = a₂; a_3it = a₃.

We understand that all the classic assumptions have to hold for us to use this pooled OLS estimator, specifically,

We all feel that the two markets cannot have the exact same characteristics most of the time. Prof. Metric says that in this case, stacking two or more datasets for different identities will bias the coefficient estimates, and the variances will be inflated, so the tests will be invalid. He tells us that the most general case of panel data is written as:

Booka exclaims, “Oh… I now see that each of the parameters a_1it, a_2it, and a_3it has the subscript it.” Prof. Metric, says, “Yes, this model allows for each identity to be different across sections and over time.” He continues by stating that we cannot estimate this model because there are not enough data points to cover all the unknown parameters. Hence, some simplifications are needed. The first way is to allow the identities to be different in their intercepts:

Prof. Metric reminds us to note the subscript i as in a_1i, which indicates different intercepts across the identities; whereas, the variables a₂ and a₃ do not have this subscript, implying that the identities have the same slope. The model in equation (6.4) assumes that all behavioral differences among the identities are captured by the constant term, so that:

a_1it = a_1i; a_2it = a₂; a_3it = a₃.

Another case occurs when sectional individuals have different slopes:

The third case combines equations (6.4) and (6.5):

These differences in characteristics are unobserved effects that need to be removed. Panel data techniques can be applied for either case, and more cases will be discussed later.

Panel Data Techniques

We first focus on the model in equation (6.4) using either the first-difference estimation or the fixed-effects estimation.

First-Difference Estimation

Given the model in equation (6.4), we can add a second equation by retrogressing one period:

Subtract the second equation from the first:

The model in equation (6.7) will control for the difference in the intercepts because the constant term in equation (6.4) has been eliminated and no longer exists in equation (6.7).

Prof. Metric then tells us that the suitable case for using a first-difference model, instead of the fixed-effects model, is when the error term follows a random walk (Wooldridge 2013).

y_it = a_1i + a₂x_2it + a₃x_3it + e_it,

where e_it = e_i,t−1 + v_it,

where v_t satisfies all the classic assumptions:

Taking the first difference yields:

In this case, taking the first difference serves two purposes: (i) elimination of the intercept a_1i presented in equation (6.4) and (ii) correcting for the autocorrelation of the errors. Since v_t satisfies all the classic assumptions, there is no longer an autocorrelation problem.

An example of the first-difference estimation is the following model:

HOUSE_it = a_1i + a₂INCOME_it + a₃CREDIT_it + e_it; e_t = e_t−1 + v_t,

where HOUSE is the average value of investment in residential housing, INCOME is per capita income, and CREDIT is the investment credit from the federal government. Going backward one period and subtracting the second equation from the first yields:

ΔHOUSE_it = a₂INCOME_it + a₃CREDIT_it + Δe_it.

Suppose that the regression results are:

ΔHOUSE_it = 1.2 ΔINCOME_it + 0.3 ΔCREDIT_it,

where ∆INCOME_it = $8,000 and ΔCREDIT_it = $ 5,000, then the point prediction of the investment in housing is:

ΔHOUSE_it = 1.2 * 4,000 + 0.3 * 5,000 = 9,600 + 1,500 = $ 11,100.

Since this is a change in the investment value instead of the investment value itself, we can calculate the predicted value of the investment as follows:

ΔHOUSE_it = HOUSE_it − HOUSE_i,t−1,

HOUSE_it = ΔHOUSE_it + HOUSE_i,t−1.

Suppose data on the previous period provides the average investment value as HOUSE_i,t−1 = $150,000, then the predicted value is:

HOUSE_it = $11,100 + $150,000 = $161,100.

We learn that the concept of differencing can be extended to more than two periods. For example, a three-period difference model will be as follows:

Δy_i,t = a₂Δx_2i,t + a₃Δx_3i,t + a₄Δx_3i,t−1 + Δe_i,t + Δe_i,t−1.

Prof. Metric tells us that in this case, a comma is often placed between the subscript for the identity and the time period to avoid any confusion on the meaning of the notation.

Fixed-Effects Estimation

Next, Prof. Metric shows us another method of controlling for the differences in the intercepts by using the “fixed effects” estimation. We again zero in on the difference in intercepts first. The theoretical model is obtained by taking the deviation from the mean, so we have to take the time average values of equation (6.4):

We then subtract equation (6.8) from equation (6.4):

We now can perform forecasts on the following model using OLS:

where

The transformed model in (6.9) will control the fixed effects problem because the intercept term in equation (6.4) has been removed from equation (6.9). The data are said to be “demeaned” because we take the deviation from the mean of the data.

In practice, the most convenient and flexible way to carry out regression on a fixed-effects model is to use the least square dummy variable (LSDV) method whenever the number of identities is not too large (less than 100 identities or time periods). To obtain the LSDV estimators for equation (6.4), we generate a dummy variable for each of the identities. Suppose we have six different identities, then:

With these six dummies added, equation (6.4) can be written as:

Prof. Metric reminds us to suppress the constant, because the model (eq. 6.9) no longer has a constant.

Prof. Metric continues with the example of pajama sales in Korea and China and gives us this model:

SALE_it = a_1i + a₂PRICE_it + a₃PROM_it + e_it,

where SALE is the values of pajama sales, PRICE is price of pajamas, and PROM is the expenditures on sale promotion. He asks us to write a model to control for the difference in sale characteristics in the two countries.

We decide to use the LSDV model and see that only the intercept term is different between the two countries, as indicated by the subscript i, so we add two intercept dummies: D_K = Korea and D_C = China and write the model as:

We also tell Prof. Metric that we will have to suppress the constant when running the regression, and he is very pleased that we remembered the details of this method.

Touro then asks, “We learned in chapter 4 that a researcher can only add (G−1) dummies. Why do we have to use two dummies for two groups here?” Prof. Metric commends Touro on the question and says that we already suppressed the constant, so there is no perfect collinearity here, and the reason we want to add two dummies is that we want to control for different characteristics in both countries.

He then says that if the two markets also differ over time, then adding time dummies to the equation will help:

SALE_it = a₁₁D_K + a₁₂D_C + a₂PRICE_it + a₃PROM_it + b₁t₁ + ... + b_Tt_T + e_it

Invo asks, “Can we add the slopes in addition to the time dummies? Or had we better stick to the model in (6.6)?” Prof. Metric says, “We can add the slope dummies to the equation, so using either cross-sectional or a combination of cross-sectional and time dummies is fine.” Combining the two produces this model:

SALE_it = a₁₁D_K + a₁₂D_C + a₂PRICE_it + a₃PROM_it + b₁t₁ + ... + b_Tt_T
+ c₁ (D_K * PRICE_it) + c₂ (D_C * PRICE_it) + d₁ (D_K * PROM_it)
+ d₂ (D_C * PROM_it) + e_it

Prof. Metric also tells us that we will learn more applications of time dummies and slope dummies in later chapters.

Seemingly Unrelated Regressions (SUR)

Because this method requires econometric software packages other than Excel, Prof. Metric only provides us with a quick introduction. Suppose that we have three equations for Singapore (1), Myanmar (2), and Laos (3). SUR estimations assume that the errors of these three equations exhibit contemporaneous correlation in the same period. The basic SUR estimation, which is a GLS procedure, can be performed in three steps, as follows:

This method is usually very effective for identities of the same region (Southeast Asia in this case), country, state, or city, because they are often correlated with each other.

Detecting Different Characteristics

Since pooled OLS can be performed when all identities or time periods have the same characteristics, we need to perform a test on these characteristics. For example, we want to know if the equations for Korea and China have identical parameters. This is an F-test, called the Chow test, for the significance of the dummy variable. The restricted model is:

SALE_it = a₁₁ + a₂PRICE_it + a₃PROM_it + e_it.

For the purpose of testing the preceding equation, the unrestricted model needs only one dummy added because a₁₁ already catches characteristics of one of the two countries, those of Korea in this case, so the equation is:

SALE_it = a₁₁ + a₁₂D_C + a₂PRICE_it + a₃PROM_it + e_it.

Suppose a₁₂ = 0, then China shares the same characteristics with Korea, so the hypotheses are written as:

H₀:a₁₂ = 0; H_a:a₁₂ ≠ 0.

Assuming that heteroscedasticity or serial correlation is not a problem with this model, the formula for F-statistics is similar to the one discussed in chapters 3 and 4, with the same definitions for J and K, except that we have a panel data, so:

where IT is the sample size with I = the number of observations across identities and T = the number of observations over time.

Prof. Metric reminds us that we still have to perform the test in the four standard steps. If the F-statistic is greater than F-critical, then we reject H₀, and the two equations do not have identical coefficients, so a panel-data technique is needed for estimations.

Goodness-of-Fit

Prof. Metric tells us that eliminating the constant amounts to regressing through the origin. This procedure causes the R² value to become an unreliable measure for the goodness-of-fit, which is also compromised for several other models in Volume Two. For this reason, he introduces the Root Mean Squared Error (RMSE) here so that we know an alternative measure for the goodness-of-fit:

We then work on an RMSE example given the information in Table 6.1.

We then calculate the mean squared errors (MSE):

MSE = (1 + 2 + 4)/3 = 2.33.

Finally, we take the square root of the MSE:

Prof. Metric says that the smaller the RMSE, the better fit a model is.

Table 6.1 Steps for calculating RMSE

(1)	(2)	(3)	(4)	(5)
Observation	y		y −	(y − )2
1	8	7	8 − 7 = 1	12 = 1
2	9	8	9 − 8 = 1	12 = 2
3	7	9	7 − 9 = − 2	(− 2)2 = 4

Data Analyses

Taila shares with us a yearly dataset on output per worker (OUT) and exports (EXPS) for Malaysia, Australia, and Cambodia during the years 2007 to 2015. She has downloaded the data from the World Bank website. Since one lagged variable is generated, we have data for the years 2008 to 2015 to perform the regressions and tests. We find that the data are in the file Ch06.xls, Fig. 6.1.

Testing Different Characteristics

The restricted model is:

EXPS_it = a₁₁ + a₂OUT_i,t−1 + e_it.

The unrestricted model is:

D_A and D_C are the dummies for Australia and Cambodia, respectively. Prof. Empirie reminds us that the three countries will share the same characteristics if a_1A = a_1C = 0, so we only need two dummies for the test.

To perform the cross-equation test on the three countries, we first estimate the restricted model by regressing EXPS on OUT:

Next, we regress EXPS on OUT, D_A and D_C:

The Analysis of Variance (ANOVA) sections that report the SSEs for the two models are displayed in Figure 6.1. We find that SEE_R is reported in cell X4 and SSE_U in cell AC4.

From the results in this figure, the four steps for the test are:

Estimating with Panel Data

First-Difference Estimation

We find that the data are in the file Ch06.xls, Fig. 6.2. The model is

ΔEXPS_it = ΔOUT_i,t−1 + e_it.

We have to perform the following steps:

Figure 6.1 ANOVA sections for restricted and unrestricted models, respectively

Figure 6.2 shows that the results for the intercept are suppressed and reported as #N/A.

Prof. Empirie points out that we were estimating the change of the variables, so we need to follow the theoretical equation to recover the intercept for predictions:

y_it = a_1i + a₂x_2it + a₃x_3it,

so,

Figure 6.2 Results for first-difference estimation

Once the intercept is recovered, we can substitute it into the estimated equation to calculate the point and interval predictions as usual.

Fixed-Effects Estimation

We find that the data are in the file Ch06.xls, Fig. 6.3. Prof. Empirie reminds us that theoretically, we can perform a fixed-effects model by using the demeaned model. Empirically, it is much more convenient to use the LSDV techniques, so we are going to perform an LSDV estimation, and the regression equation is:

EXPS_it = a_1AD_A + a_1CD_C + a_1MD_M + OUT_i,t−1 + e_it.

We are sure that you noticed the three dummies added to the equation and that the constant is suppressed from this equation. You should perform the following regression steps:

The results are reported in Figure 6.3. You can see that we use all three dummies to control for the fixed effects.

Prof. Empirie reminds us that recovering the intercept is possible by using the model:

so,

Figure 6.3 Results for fixed-effects estimation

Once the intercept is recovered, substitute it into the estimated equation to calculate the point and interval predictions as usual.

Exercises