3.2 An econometric model for the study
The study analyzes the impact of terms of trade (TOT), labor, and capital, on the United States’ economic growth. This study has followed similar models as those used by (Jebran et al., 2018a; Nancy, 2021) in their studies. To use GPD as a representation of economic growth and term of the trade (TOT) as the fraction of exports and imports. GDP as a function of TOT is written as follows Eq. 1.
\({GDP}_{t}\) = \(f(Labor, Capital, TOT)\) Eq. 1
Where; GDPt represents the Gross Domestic Product over a time period t, and term of the trade (TOT) represents the exports and imports ratio over the time period t. Capital represents the total gross capital and labor denotes the total labor force of the United States.
The above Eq. 1 can be expressed in Eq. 2 for a log-linear model or short-run and long-run models to scrutinize the association between GDP and TOT, labor, and Capital.
\({LogGDP}_{t}\) = \({\beta }_{0}\)+\({\beta }_{1}Log{TOT}_{t}\)+ \({\beta }_{2}Log{Capital}_{t}\) + \({\beta }_{3}Log{Labor}_{t}\) +\(\mu\) Eq. 2
Where; \({LogGDP}_{t}\) is the log of GDP over time t; \({\beta }_{0}\)= constant; \({\beta }_{1}\), \({\beta }_{2}\) and \({\beta }_{3}\) = coefficient of the ith independent variables and \({TOT}_{t}\), \({Capital}_{t}\) and \({Labor}_{t}\) = independent variable; and \(\mu\)=error term.
For estimation of long-term dynamics, the linear model can be further expanded into error correction model Eq. 3 as follows.
\({LogGDP}_{t}\) = \({\varPhi }_{1i}\)\(\sum _{i=0}^{q}{\varDelta LogGDP}_{t-i}\) + \({\varPhi }_{2i}\sum _{i=0}^{q}\varDelta Log{TOT}_{t-1}\)+ \({\varPhi }_{3i}\sum _{i=0}^{q}\varDelta Log{Capital}_{t-1}\) + \({\varPhi }_{4i}\sum _{i=0}^{q}\varDelta Log{Labor}_{t-1}\) + \(\mu\) Eq. 3
The short-run relationship model can be shown in Eq. 4 as follows.
\(LogGDP\) = \({\psi }_{1i}\)\(\sum _{i=0}^{q}{\varDelta LogGDP}_{t-i}\) + \({\psi }_{2i}\sum _{i=0}^{q}\varDelta Log{TOT}_{t-1}\)+ \({\psi }_{3i}\sum _{i=0}^{q}\varDelta Log{Capital}_{t-1}\) + \({\psi }_{4i}\sum _{i=0}^{q}\varDelta Log{Labor}_{t-1}\) + \(\mu\) Eq. 4
In both the above Eq. 3 and Eq. 4, the other variables are the same as in Eq. 2 except \({LogGDP}_{t-i}\) which represents the Gross Domestic Product of one lagged period error correction term. Furthermore, \(Log{TOT}_{t-1}\)shows the one lagged value of error correction terms of trade (TOT), \(Log{Capital}_{t-1}\)demonstrations of the one-lagged error correction value of Capital and \(Log{Labor}_{t-1}\)shows the one lagged error correction value of Labor, while \(\varDelta\) is first difference value and \({\varPhi }_{1i}\), \({\varPhi }_{2i}\), \({\varPhi }_{3i}\) and \({\varPhi }_{4i}\) explain the long-run coefficient in Eq. 3. However, \({\psi }_{1i}\), \({\psi }_{2i}\), \({\psi }_{3i}\) and \({\psi }_{4i }\)explain the short-run coefficient in Eq. 4. The same modeling was applied in the previous literature as reported by (Jebran et al., 2018a).
We utilized the Augmented Dickey-Fuller (ADF) test, which is common in the literature, to find whether the time series data is stationary or not (Jebran et al., 2018b). Dickey and Fuller (1979) give a new concept to test autoregressive related to time series data which is called the ADF unit root test. In 1984, they developed the Dickey-Fuller test to make it easier to test the autoregressive unit root test constructed on the Dickey-Fuller test on a time series sample. The ADF model comprises both null and alternative hypotheses, thus we need to see the p-value presented as a guide to whether the data we are using in a study is stationary or non-stationary in order to check the hypothesis results. However, Peter and Perron (1988) also established Philipps-Perron Test (PP-Test) for unit root to test the serial correlation between variables. Hence, Unit root test Eq. 5 is mentioned below:
\({P}_{t}\) = \({\beta }_{0}{P}_{t-1}\) + \({\beta }_{1}{X}_{e}\) + \(\varPhi\) Eq. 5
We are going to test a time series value \({P}_{t}\)during the period of t, where \({\beta }_{0}\) and \({\beta }_{1}\) are explanatory variables, \({X}_{e}\) is represents an exogenous variable and \(\varPhi\) is represents a model’s error term.
The Augmented Dickey-Fuller (ADF) model is a unit root test Eq. 6, which explains the null hypothesis \({\alpha }_{0}\) = 1 that takes the coefficient of lag(1) of\({Y}_{t}\) .
Null Hypothesis: \({\alpha }_{0}\) = 1
\({Y}_{t}\) = c + \({\beta }_{0}\) + \({\alpha }_{0}{y}_{t-1}\) + \(\psi \varDelta {Y}_{t-1}\) + \(\mu\) Eq. 6
Where; \({Y}_{t}\) is a time series value during the period of t; \({\alpha }_{0}\) and \(\psi\) are explanatory variables; \({y}_{t-1}\) is representing time series first lag; \(\varDelta {Y}_{t-1}\) shows representing time series at the time (t-1) first difference lag.
For the high-order regression model, the above Eq. 6, ADF unit root test can be modified into Eq. 7 as follows.
\({P}_{t}\) = c + \({\beta }_{0}\) + \({\alpha }_{0}{y}_{t-1}\) + \({ \text{ϸ}}_{1}\varDelta {P}_{t-1}\) + \({ \text{ϸ}}_{2}\varDelta {P}_{t-2}\) + \({ \text{ϸ}}_{3}\varDelta {P}_{t-3}\) ..…. +\({ \text{ϸ}}_{p}\varDelta {P}_{t-n}\) + \(ϵ\)Eq. 7
Here, we have added just more different terms which are\(\varDelta {Y}_{t-1}\), \(\varDelta {P}_{t-2}\), \(\varDelta {P}_{t-3}\) and \(\varDelta {P}_{t-n}\) respectively.
To further scrutinized our data, Engle and Granger (1987) gave a concept of testing for co-integration which combines the problem of test of unit root and tests with parameters unknown under \({H}_{0}\). In addition, Johansen and Juselius (1990) developed the ARDL model to test the co-integration analysis between two or more variables. Wong (2010) and Pesaran et al. (2001) have used the ARDL bound test models in their studies to test the short and long-run impact. The models applied in our study are the Augmented Dickey-Fuller (ADF) and ARDL models to test the short and long-run relationship between the United State economic growth and term of trade TOT, Labor, and Capital. However, the Johansen test is an improved form of Clive Granger and the Phillips-Ouliaris tests Johansen test has two subtests as Trace test and the Maximum test.
The trace test, which is the first stage of the Johansen test, may calculate the co-integration of the data from various time series. One of two ideas can account for the trace test results:
\({H}_{0}\) = Variable K must be less or equal to m-1
\({H}_{1}\) = Variable K must be greater or equal to m-1
The following two assumptions are mentioned in the Maximum Eigenvalue test, the second step of the Johansen test, along with details on the co-integration of several time series:
\({H}_{0}\) = K has co-integration
\({H}_{1}\) = K + 1 has co-integration