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Constructing a Statistical Model of Per-Capita GDP Change in Q2 2020 Using GRETL

## Introduction to GRETL - A Free Statistical Analysis Software Package

The GDP data from Problem Set 1b can be used to construct a statistical model of the change in per-capita GDP for second quarter (April to June) 2020. We will use the information for the 33 countries in our data set to examine the factors that influenced the magnitude of the COVIDinduced recession. Although professional economists use very sophisticated models of GDP, we
can apply a few basic tools to construct a simple but effective model. Our approach is based on a statistical tool known as Ordinary Least Squares (OLS) which is used to construct a linear regression model. Please load the data set into GRETL based on the instructions for Problem Set 1b and then answer the following questions:

1. First, we estimate a base model of the percentage change in GDP per capita, and the variables were defined in Problem Set 1b:

Q2GDP = b1 + b2*BB + b3*BUSWEB + b4*SOCEXP + b5*CHILD + b6*SERVICE

In words, this regression model states that the expected percentage change in GDP per capita (Q2GDP) for the 33 countries in our sample may be explained by five factors (BB, BUSWEB, CHILD, TOUR, and SERVICE). To understand the role of these model components, we can examine the second term on the right side of the equal sign. Here, the expected percentage change in GDP
per capita depends on the share of households with access to high-speed internet access (BB), and the coefficient b2 represents the expected percentage change in GDP per capita if the share of households with high-speed internet access increases by one unit (e.g., from 50% to 51%) and we hold all other variables constant.

In general, we expect internet access to promote economic growth, so we expect the estimated value of b2 to be positive. The other coefficients to the right of this term (b3 to b6) can be interpreted in similar fashion. Finally, b1 represents the expected
percentage change in GDP per capita if all of the other explanatory factors on the right side of the equation equal zero. This situation is unlikely, so there is not a nature interpretation for b1.

We can use the Ordinary Least Squares (OLS) tool in GRETL to estimate the model coefficients (b1-b6) from the data set. OLS is widely used in regression analysis because it has some very favorable statistical properties. In particular, the estimator is unbiased – across data sets drawn from the same source, the expected value of the estimated coefficients equals the true value of the coefficients. Also, OLS is efficient, which implies that this estimator has smaller variance than any other unbiased estimator of the model coefficients.

## Download Latest GRETL Version and Install on Your System

To compute the OLS estimates of the model coefficients in GRETL, please click on Model in the top menu and then select Ordinary Least Squares from the submenu. The pop-up menu allows you to select the dependent variable (Q2GDP) and the explanatory variables (BB, BUSWEB, CHILD, TOUR, and SERVICE) for our base model, and the resulting window should look like:

Click on OK to compute the ordinary least square (OLS) estimates of the base model, and GRETL will provide a new pop-up window that should look like:

Please provide a copy of your regression output to verify that you got the same results. Screen shots are acceptable, and your GRETL output should not have the blue highlighted region above.

2. After estimating a regression model, we should evaluate how well the model fits the observed GDP growth data. The key statistics are highlighted with blue in the screenshot above:

a) The R-squared statistic measures the percent of the variation in the dependent variable (Q2GDP) that is explained by the fitted regression model, so higher values are preferred. For example, R2 = 0.50 indicates that the fitted model explains 50% of the variation in real GDP growth. As long as the model includes an intercept or constant term (const), the R-squared statistic takes values between zero and one. We can always increase R-squared by adding new variables to the model, even if they do not have any reasonable relationship to Q2GDP. The adjusted R-squared statistic accounts for this drawback of R-squared, and its value only increases if a new variable has a substantive impact on the model fit. Please state how we should interpret the R-squared statistic (0.614823) from this base model.

b) The F statistic can be used to test the null hypothesis that all of the explanatory variables (except const) have regression coefficients equal to zero (i.e., the model has no explanatory power for Q2GDP). If this is not true, then the model has some explanatory power for Q2GDP and we should reject the null hypothesis. The p-value measures the strength of the evidence,
and we should reject the null hypothesis if the observed p-value is less than 0.10. Note that GRETL expresses very small numbers in scientific notation. For example, a p-value stated as 1.14e-10 equals 1.14 X 10-10 or 0.000000000114. For this model, GRETL does not have to use scientific notation, and the stated p-value is p = 0.000056. Based on the observed evidence from the base model, what do we conclude from the F-test statistic?

## Use GDP Data Set from Problem Set 1b to Construct Statistical Model using Ordinary Least Squares (OLS)

3. The estimated coefficient for BB is b2 = 0.0923, which is positive as expected. In words, this estimated value implies that a unit increase in the share of households with high-speed internet access increases GDP growth per capita by 0.0923% (about one-tenth of one percent). The F-test discussed in Question 2b evaluates the overall fit of the model. To examine the statistical significance of the individual model components, we can use the t-ratio and p-value associated with each estimated coefficient. For example, the t-ratio for BB is 4.416 with p-value 0.0001. Under the null hypothesis for each explanatory variable, the coefficient equals zero and does not affect GDP growth. The t-ratio increases in absolute value and the p-value declines toward zero as the strength of the evidence against the null hypothesis increases. We should reject the null hypothesis and conclude that the coefficient is significantly non-zero if the p-value is less than 0.10. Otherwise, if the p-value is larger than 0.10, we should fail to reject the null hypothesis and conclude that the coefficient is not significantly different from zero. Here, the p-value for BB is less than 0.10, so BB is a statistically significant explanatory variable for Q2GDP. Based on the pvalues in the GRETL output, is the coefficient associated with CHILD significantly different from zero (i.e., p-values less than 0.10)? Please briefly explain how we should interpret the estimated value of the CHILD coefficient.

For the research component of this class, we will use a free statistical analysis software package known as GRETL. This package is relatively easy to use and offers most of the tools required for the research paper that you will complete for this class. Also, GRETL is available for MAC and PC operating systems. For Problem Set 1b, please complete the following tasks:

1. The latest version of GRETL is numbered 2021d that was built on September 30, 2021. For Windows users, go to gretl.sourceforge.net/win32/ and download the version of the program that works for you (32-bit or 64-bit EXE file). For MAC users, go to gretl.sourceforge.net/osx.html and download the version of the program that works for you (depending on the vintage of your CPU). Please install the software on your Windows or MAC computer.

2. The objective of the term project is to examine the factors that influenced the sharp decline gross domestic product (GDP) that most countries experienced in second-quarter 2020 due to the COVID-induced closures. To accomplish this task, we will build a statistical model of the observed changes in second-quarter GDP conditional on several explanatory variables. To begin, please download the posted Excel file (OECD data 2020.xlsx), which includes annual observations for n=33 countries:

• Country observation identifier

• Q2GDP percentage change in real GDP per capita from the previous quarter

• BB share of household that have access to high-speed broadband internet

• Busweb share of businesses that have webpages

• Socexp share of government expenditures allocated to social programs

• Child share of children under age 5 with access to childcare or preschool

• Tour – tourism industry share of total employment

• Service – share of GDP produced by the service sector of the economy

3. Open GRETL and click on the FILE menu. The first menu item should be Open data, and you should click on this link as well as the User data... item on the sub-menu. To load the OECD data set, you should direct GRETL to the directory where you saved it. Please note that you may have to change the file suffice to XLSX with the option box provided in the lower-right corner of the open window.

• After you ask gretl to open the file, you should see a pop-up window that asks if you should start importing the data from Row 1 and Column 1 of the Excel file. You should click on OK in this window.

• Next, GRETL will show a pop-up window that states the imported data are interpreted as cross-sectional observations. gretl asks if you want to change this character of the data, and you should click on no.