Box-Pierce Test of autocorrelation in Panel Data using Stata.

The test of Box & Pierce was derived from the article “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models” in the Journal of the American Statistical Association (Box & Pierce, 1970).

The approach is used to test first-order serial correlation, the general form of the test is given the statistic as:

Where the statistic of Box- Pierce Q is defined as the product between the number of observations and the sum of the square autocorrelation ρ in the sample at lag h. The test is closely related to the Ljung & Box (1978) autocorrelation test, and it used to determine the existence of serial correlation in the time series analysis. The test works with chi-square distribution by the way.

The null hypothesis of this test can be defined as H0: Data is distributed independently, against the alternative hypothesis of H1: Data is not distributed independently. Therefore, the null hypothesis is that data is not suffering from an autocorrelation structure against the alternative which proposes that the data has an autocorrelation structure.

The test was implemented in Stata with the panel data structure by Emad Abd Elmessih Shehata & Sahra Khaleel A. Mickaiel (2004), the test works in the context of ordinary least squares panel data regression (the pooled OLS model). And we will develop an example here.

First we install the package using the command ssc install as follows:

ssc install lmabpxt, replace

Then we will type help options.

help lmabpxt

From that we got the next result displayed.

We can notice that the sintax of the general form is:

lmabpxt depvar indepvars [if] [in] [weight] , id(var) it(var) [noconstant coll ]

In this case id(var) and it(var) represents the identificatory of individuals (id) and identificatory of the time structure (it), so we need to place them in the model.

Consider the next example

clear all
use http://www.stata-press.com/data/r9/airacc.dta
xtset airline time,y
reg pmiles inprog
lmabpxt  pmiles inprog, id(airline) it(time)

Notice that the Box-Pierce test implemented by Emad Abd Elmessih Shehata & Sahra Khaleel A. Mickaiel (2004) will re-estimate the pooled regression. And the general output would display this:

In this case, we can see a p-value associated to the Lagrange multiplier test of the box-pierce test, and such p-value is around 0.96, therefore, with a 5% level of significance, we cannot reject the null hypothesis, which is the No AR(1) panel autocorrelation in the residuals.

Consider now, that you might be using fixed effects approach. A numerical approach would be to include dummy variables (in the context of least squares dummy variables) of the individuals (airlines in this case) and then compare the results.

To do that we can use:

tab airlines, gen(a)

and then include from a2 to a20 in the regression structure, with the following code:

lmabpxt  pmiles inprog a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 , id(airline) it(time)

This would be different from the error component structure, and it would be just a fixed effects approach using least squares dummy variable regression. Notice the output.

Using the fixed effects approach with dummy variables, the p-value has decreased significantly, in this case, we reject the null hypothesis at a 5% level of significance, meaning that we might have a problem of first-order serial correlation in the panel data.

With this example, we have done the Box-Price test for panel data (and additionally, we established that it’s sensitive to the fixed effects in the regression structure).

Notes:

The lmabpxt appears to be somewhat sensitive if the number of observations is too large (bigger than 5000 units).

There are an incredible compilation and contributions made by Shehata, Emad Abd Elmessih & Sahra Khaleel A. Mickaiel which can be found in the next link:

http://www.haghish.com/statistics/stata-blog/stata-programming/ssc_stata_package_list.php

I suggest you to check it out if you need anything related to Stata.

Bibliography

Box, G. E. P. and Pierce, D. A. (1970) “Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models”, Journal of the American Statistical Association, 65: 1509–1526. JSTOR 2284333

G. M. Ljung; G. E. P. Box (1978). “On a Measure of a Lack of Fit in Time Series Models”. Biometrika 65 (2): 297-303. doi:10.1093/biomet/65.2.297.

Shehata, Emad Abd Elmessih & Sahra Khaleel A. Mickaiel (2014) LMABPXT: “Stata Module to Compute Panel Data Autocorrelation Box-Pierce Test”

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Ramsey RESET Test on Panel Data using Stata

In regression analysis, we often check the assumptions of the econometrical model regressed, during this, one of the key assumptions is that the model has no omitted variables (and it’s correctly specified). In 1969, Ramsey (1969) developed an omitted variable test, which basically uses the powers of the predicted values of the dependent variable to check if the model has an omitted variable problem.

Assume a basic fitted model given by:

Where y is the vector of containing the dependent variable with nx1 observations, X is the matrix that contains the explanatory variables which is nxk (n are the total observations and k are the number of independent variables). The vector b represents the estimated coefficient vector.

Ramsey test fits a regression model of the type

Where z represents the powers of the fitted values of y, the Ramsey test performs a standard F test of t=0 and the default setting is considering the powers as:

In Stata this is easily done with the command

estat ovtest

after the regression command reg.

To illustrate this, consider the following code:

use https://www.stata-press.com/data/r16/auto
regress mpg weight foreign
estat ovtest

The null hypothesis is that t=0 so it means that the powers of the fitted values have no relationship which serves to explain the dependent variable y, meaning that the model has no omitted variables. The alternative hypothesis is that the model is suffering from an omitted variable problem.

In the panel data structure where we have multiple time series data points and multiple observations for each time point, in this case we fit a model like:

With i=1, 2, 3, …, n observations, and for each i, we have t=1, 2, …, T time periods of time. And v represents the heterogenous effect which can be estimated as parameter (in fixed effects: which can be correlated to the explanatory variables) and as variable (in random effects which is not correlated with the explanatory variables).

To implement the Ramsey test manually in this regression structure in Stata, we will follow Santos Silva (2016) recommendation, and we will start predicting the fitted values of the regression (with the heterogenous effects too!). Then we will generate the powers of the fitted values and include them in the regression in (4) with clustered standard errors. Finally, we will perform a significant test jointly for the coefficients of the powers.

use https://www.stata-press.com/data/r16/nlswork

xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure c.tenure#c.tenure 2.race not_smsa south, fe cluster(idcode)

predict y_hat,xbu

gen y_h_2=y_haty_hat gen y_h_3=y_h_2y_hat

gen y_h_4=y_h_3*y_hat

xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure c.tenure#c.tenure 2.race not_smsa south y_h_2 y_h_3 y_h_4, fe cluster (idcode)

test y_h_2 y_h_3 y_h_4

Alternative you can skip the generation of the powers and apply them directly using c. and # operators in the command as it follows this other code:

use https://www.stata-press.com/data/r16/nlswork

xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure c.tenure#c.tenure 2.race not_smsa south, fe cluster(idcode)

predict y_hat,xbu

xtreg ln_w grade age c.age#c.age ttl_exp c.ttl_exp#c.ttl_exp tenure c.tenure#c.tenure 2.race not_smsa south c.y_hat#c.y_hat c.y_hat#c.y_hat# c.y_hat c.y_hat#c.y_hat# c.y_hat# c.y_hat , fe cluster (idcode)

test c.y_hat#c.y_hat c.y_hat#c.y_hat# c.y_hat c.y_hat#c.y_hat# c.y_hat# c.y_hat

At the end of the procedure you will have this result.

Where the null hypothesis is that the model is correctly specified and has no omitted variables, however in this case, we reject the null hypothesis with a 5% level of significance, meaning that our model has omitted variables.

As an alternative but somewhat more restricted, also with more features, you can use the user-written package “resetxt” developed by Emad Abd & Sahra Khaleel (2015) which can be used after installing it with:

ssc install resetxt, replace

This package however doesn’t work with factor-variables or time series operators, so we cannot include c. or i. and d. or L. operators for example.

clear all

use https://www.stata-press.com/data/r16/nlswork

gen age_sq=ageage gen ttl_sq= ttl_exp ttl_exp

gen tenure_sq= tenure* tenure

xtreg ln_w grade age age_sq ttl_exp ttl_sq tenure tenure_sq race not_smsa south, fe cluster(idcode)

resetxt ln_w grade age age_sq ttl_exp ttl_sq tenure tenure_sq race not_smsa south, model(xtfe) id(idcode) it(year)

however, the above code might be complicated to calculate in Stata, depending on how much memory do you have to do the procedure. That’s why in this post it was implemented the manual procedure of the Ramsey test in the panel data structure.

Bibliography

Emad Abd, S. E., & Sahra Khaleel, A. M. (2015). RESETXT: Stata Module to Compute Panel Data REgression Specification Error Tests (RESET). Obtained from: Statistical Software Components S458101: https://ideas.repec.org/c/boc/bocode/s458101.html

Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society Series B 31, 350–371.

Santos Silva, J. (2016). Reset test after xtreg & xi:reg . Obtained from: The Stata Forum: https://www.statalist.org/forums/forum/general-stata-discussion/general/1327362-reset-test-after-xtreg-xi-reg?fbclid=IwAR1vdUDn592W6rhsVdyqN2vqFKQgaYvGvJb0L2idZlG8wOYsr-eb8JFRsiA

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A brief example to model the Cobb-Douglas utility function using Stata.

Regarding microeconometrics, we can find applications that go from latent variables to model market decisions (like logit and probit models) and techniques to estimate the basic approaches for consumers and producers.

In this article, I want to start with an introduction of a basic concept in microeconomics, which is the Cobb-Douglas utility function and its estimation with Stata. So we’re reviewing the basic utility function, some mathematical forms to estimate it and finally, we’ll see an application using Stata.

Let’s start with the traditional Cobb-Douglas function:

Depending on the elasticity α and β for goods X and Y, we’ll have a respective preference of the consumer given by the utility function just above. In basic terms, we restrict α + β =1 in order to have an appropriate utility function which reflects a rate of substitution between the two goods X and Y.  If we assume a constant value of the utility given by U* for the consumer, we could graph the curve by solving the equation for Y, in this order of ideas.

And the behavior of the utility function will be given by the number of quantities of the good Y explained by X and the respective elasticities α and β. We can graph the behavior of the indifference curve given a constant utility level according to the quantities of X and Y, also for a start, we will assume that α =0.5 and β=0.5 where the function has the following pattern for the same U* level of utility (example U=10), this reflects the substitution between the goods.

Cobb-Douglas function with U=10 and α =0.5 and β=0.5 . Source: Own Elaboration

If you might wonder what happens when we alter the elasticity of each good, like for example, α =0.7 and β=0.3 the result would be a fast decaying curve instead of the pattern of the utility before.

Cobb-Douglas function with U=10 and α =0.7 and β=0.3 . Source: Own Elaboration

Estimating the utility function of the Cobb-Douglas type will require data of a set of goods (X and Y in this case) and the utility.

Also, it would imply that you somehow measured the utility  (that is, selecting a unit or a measure for the utility), sometimes this can be in monetary units or more complex ideas deriving from subjective utility measures.

Applying logarithms to the equation of the Cobb-Douglas function would result in:

Which with properties of logarithms can be expressed as:

This allows a linearization of the function as well, and we can see that the only thing we don’t know regarding the original function is the elasticities of α and β. The above equation fits perfectly in terms of a bivariate regression model. But remember to add the stochastic part when you’re modeling the function (that is, including the residual in the expression). With this, we can start to do a regressing exercise of the logarithm of the utility for the consumers taking into account the amount of the demanded goods X and Y. The result would allow us to estimate the behavior of the curve.

However, some assumptions must be noted: 1) We’re assuming that our sample (or subsample) containing the set of individuals i tend to have a similar utility function, 2) the estimation of the elasticity for each good, would also be a generalization of the individual behavior as an aggregate. One could argue that each individual i has a different utility function to maximize, and also that the elasticities for each good are different across individuals. But we can argue also that if the individuals i  are somewhat homogenous (regarding income, tastes, and priorities, for example, the people of the same socioeconomic stratum) we might be able to proceed with the estimation of the function to model the consumer behavior toward the goods.

The Stata application

As a first step would be to inspect the data in graphical terms, scatter command, in this case, would be useful since it displays the behavior and correlation of the utility (U) and the goods (X and Y), adding some simple fitting lines the result would be displayed like this:

twoway scatter U x || lfit U x
twoway scatter U y || lfit U y 
Stata graphs for the dispersion of each good (X and Y) relative to the utility. Source: Own Elaboration.

Up to this point, we can detect a higher dispersion regarding good Y. Also, the fitted line pattern relative to the slope is different for each good. This might lead to assume for now that the overall preference of the consumer for the n individuals is higher on average for the X good than it is for the Y good. The slope, in fact, is telling us that by an increase of one unit in the X good, there’s a serious increase in the utility (U) meanwhile, the fitted line on the good Y regarding to its slope is telling us comparatively speaking, that it doesn’t increase the utility as much as the X good. For this cross-sectional study, it also would become more useful to calculate Pearson’s correlation coefficient. This can be done with:

correlate U y x
Correlation Matrix between the variables. Source: Own Elaboration

The coefficient is indicating us that exists somewhat of a linear association between the utility (U) and the good Y, meanwhile, it exists a stronger linear relationship relative to the X good and the utility. As a final point, there’s an inverse, but not significant or important linear relationship between goods X and Y. So the sign is indicating us that they’re substitutes of each other.

Now instead of regressing U with X and Y, we need to convert it into logarithms, because we want to do a linearization of the Cobb-Douglas utility function.

gen ln_U=ln(U)
gen ln_X=ln(x)
gen ln_Y=ln(y)
reg ln_U ln_X ln_Y
Regression with Ordinary Least Squares with constant. Source: Own Elaboration

And now performing the regression without the constant.

Regression with Ordinary Least Squares without the constant. Source: Own Elaboration

Both regressions (with and without the constant) tends to establish the parameters in α =0.6 and β=0.4 which matches the Data Generating Process of the Montecarlo simulation. It appears that the model with the constant term has a lesser variance, so we shall select these parameters for further analysis.

How would it look then our estimation of this utility function for our sample? well, we can start using the mean value of the utility using descriptive statistics and then use a graphical function with the parameters associate. Remember that we got:

And we know already the parameters and also we can assume that the expected utility would be the mean utility in our sample. From this, we can use the command:

sum U y x
Summary of the variables using Stata. Source: Own Elaboration

And with this, the estimated function for the utility level U=67.89 with approximated elasticities of 0.6 and 0.4 would look like this:

Graph of the Curve for the Expect Utility of the Sample, with the parameters estimated with OLS. Source: Own Elaboration

In this order of ideas, we just estimated the indifference curve for a certain population which consists of a set of i individuals. The expected utility from both goods was assumed as the mean value of the utility for the sample and with this, we can identify the different sets of points related to the goods X and Y which represents the expected utility. This is where it ends our brief example of the modeling related to the Cobb-Douglas utility function within a sample with two goods and defined utilities.

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