Introduction

In this blog, we extend last week's analysis of unit root testing with structural breaks to panel data.

We will again use the quarterly current account to GDP ratio but focus on a panel of data from five countries: United States, United Kingdom, Australia, South Africa, and India.

Using panel data unit roots tests found in the GAUSS tspdlib library we consider if the panel collectively shows unit root behavior.

Testing for unit roots in panel data

Why panel data

There are a number of reasons we utilize panel data in econometrics (Baltagi, 2008). Panel data:

• Capture the idiosyncratic behaviors of individual groups with models like the fixed effects or random effects models.
• Contain more information, more variability, and more efficiency.
• Can detect and measure statistical effects that pure time-series or cross-section data can't.
• Provide longer time-series for unit-root testing, which in turn leads to standard asymptotic behavior.

Panel data unit root testing

Today we will test for unit roots using the panel Lagrangian Multiplier (LM) unit-root test with structural breaks in the mean (Im, K., Lee, J., Tieslau, M., 2005):

• The panel LM test statistic averages the individual LM test statistics which are computed using the pooled likelihood function.
• The asymptotic distribution of the test is robust to structural breaks.
• The test considers the null unit root hypothesis against the alternative that at least one time series in the panel is stationary.

Testing our panel

Setting up the test

The panel LM test can be run using the GAUSS PDLM procedure found in the GAUSS tspdlib library. The procedure has two required inputs and four additional optional arguments:

y_test

T x N matrix,  the panel data to be tested.

model

Scalar,  indicates the type of model to be tested.
    1 = break in level.
    2 = break in level and trend.

nbreak

Scalar,  optional input, the number of breaks to allow.
    1 = one break.
    2 = two breaks. Default = 0.

pmax

Scalar,  optional input, maximum number of lags for Dy. 0 = no lags. Default = 8.

ic

Scalar,  optional input, the information criterion used to select lags.
    1 = Akaike.
    2 = Schwarz.
    3 = t-stat significance. Default = 3.

trimm

Scalar,  optional input, data trimming rate. Default = 0.10

---

The PDLM procedure has five returns:

Nlm

Vector,  the minimum test statistic for each cross-section.

Ntb

Vector,  location of break(s) for each cross-section.

Np

Scalar,  number of lags selected by chosen information criterion for each cross-section.

PDlm

Scalar,  panel LM statistic with N(0, 1).

pval

Scalar,  p-value of PDlm.

Running the test

The test is easy to set up and run in GAUSS. We first load the tspdlib library and our data.

library tspdlib;

// Load data
ca_panel = loadd("panel_ca.dat");
y_test = ca_panel[., 2:cols(ca_panel)];

Next, we specify that we want to run the model with level breaks and we call the PDLM procedure separately for the one break and two break models. We will keep all other parameters at their default values:

// Specify to run model with 
// level breaks
model = 1;

// Run first with one break
nbreak = 1;

// Call PD LM with one level break
{ Nlm, Ntb, Np, PDlm, pval } = PDLM(y_test, model, nbreak);

// Run next with two breaks
nbreak = 2;

// Call PD LM with level break
{ Nlm, Ntb, Np, PDlm, pval } = PDLM(y_test, model, nbreak);

Results

Research on the presence of unit roots in current account balances has had mixed results. These results bring to the forefront the question of current account balance sustainability (Clower & Ito, 2012).

Our panel tests with structural breaks unanimously reject the null hypothesis of unit roots for all cross-sections, as well as the combined panel. This adds support, at least for our small sample, to the idea that current account balances are sustainable and mean-reverting.

Conclusions

Today we've learned about conducting panel data unit root testing in the presence of structural breaks using the LM test from (Im, K., Lee, J., Tieslau, M., 2005). After today you should have a better understanding of:

1. Some of the advantages of using panel-data.
2. How to test for unit roots in panel data using the LM test with structural breaks.
3. How to use the GAUSS tspdlib library to test for unit roots with structural breaks.

Code and data from this blog can be found here.

Further Reading

1. Panel Data Basics: One-way Individual Effects
2. How to Aggregate Panel Data in GAUSS
3. Introduction to the Fundamentals of Panel Data
4. Panel Data Stationarity Test With Structural Breaks
5. Transforming Panel Data to Long Form in GAUSS

References

Baltagi, B. (2008). Econometric analysis of panel data. John Wiley & Sons.

Clower, E., & Ito, H. (2012). The persistence of current account balances and its determinants: the implications for global rebalancing.

Im, K., Lee, J., Tieslau, M. (2005). Panel LM Unit-root Tests with Level Shifts. Oxford Bulletin of Economics and Statistics 67, 393–419.

Linear regression commonly assumes that the error terms of a model are independently and identically distributed (i.i.d). However, when datasets contain groups, the potential for correlated error terms within groups arises.

Example: Weather shocks to apple orchards

For example, consider a model of the supply of apples from various orchards across the United States. Naturally, we would expect that orchards within Washington may all face similar weather-related "shocks" to their supply. However, we would not expect the weather shocks to the orchards in Washington to be the same as the weather shocks to the orchards in New York.

When these correlated within-group shocks occur, the i.i.d error term assumption is not valid and traditional error terms can result in misleading inference about coefficient estimates.

In these cases, is important to use error terms that appropriately account for the within-group correlation between error terms.

The Model

Let's consider our hypothetical apple supply model which makes observations for each individual orchard, $i = 1, 2, \ldots\, N$ at each time period $t = 1, 2, \ldots\, T$:

$$y_{it} = x_{it}\beta + u_{it}$$

We can further aggregate our dataset into state-level groups, $g = 1, 2, \ldots, G$ such that:

$$y_{igt} = x_{igt}\beta + u_{igt}$$

The cluster-robust error term assumes that, $u_{igt}$, is correlated within groups but independent across groups. More formally:

$$E[u_{igt}u_{iht}] \begin{cases} = 0 & \text{ if }g \neq h \\ \neq 0 & \text{ if }g = h \end{cases} $$

The cluster-robust error computation allows for this correlation:

$$V_{clu}[\hat{\beta}] = (X'X)^{-1} * \sum_{j=1}^G u_j' u_j * (X'X)^{-1}$$

where

$$u_j = \sum_{cluster_j} u_{it} x_{it} .$$

Estimating our model in GAUSS

Let's look more formally at our apple production model using the apples_cluster.dat dataset. Using this data we will model the production of apples in relationship to orchard acreage:

$$prod = \beta_0 + \beta_1*acres + u$$

Estimating i.i.d error terms

First, let's model the data using i.i.d standard errors:

// Specify filename
fname = __FILE_DIR $+ "apples_cluster.dat";

// Estimate model using ols
struct olsmtOut oOut;
oOut = olsmt(fname, "prod ~ acres", oCtl);

This yields the following results:

                         Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------
CONSTANT    0.0296797   0.0283593     1.04656     0.295       ---         ---
acres         1.03483   0.0285833     36.2041     0.000    0.455813    0.455813 

Estimating cluster-robust error terms

Now, we specify cluster-robust errors using two members in the olsmtControl structure:

oCtl.cov

String, the type of covariance matrix to be computed:

• "iid" for i.i.d errors.
• "cluster" for cluster-robust errors.
• "robust" for the Huber/White sandwich estimator.

oCtl.clusterId

String, the name of the variable containing data groups.

Our code for estimation now becomes:

// Estimate model using ols
struct olsmtControl oCtl;
oCtl = olsmtControlCreate();

// Set up cluster id variable
oCtl.clusterId = "state";

// Turn on cluster vce
oCtl.cov = "cluster";

// Estimate model 
struct olsmtOut oOut;
oOut = olsmt(fname, "prod ~ acres", oCtl);

Which yields:

                         Standard                 Prob   Standardized  Cor with
Variable     Estimate      Error      t-value     >|t|     Estimate    Dep Var
-------------------------------------------------------------------------------
CONSTANT    0.0296797   0.0670127    0.442897     0.658       ---         ---
acres         1.03483   0.0505957      20.453     0.000    0.455813    0.455813 

Comparing results

There are several key things to note about the two sets of results.

1. Using cluster-robust standard errors has no impact on the coefficient estimates.
2. The cluster-robust standard errors are larger than i.i.d errors.

In this case, the larger standard errors do not impact our conclusions regarding the significance of the estimated coefficients, but this may not always be true.

Conclusions

In today's discussion of cluster-robust standard errors we have learned :

1. What types of models may introduce within-cluster correlation in error terms.
2. The potential impacts of ignoring within-cluster correlations in error terms.
3. How to estimate cluster-robust error terms.

Code and data from this blog can be found here.