Question

In Example 10.4 , we wrote the model that explicitly contains the long-run propensity, \(\theta_{0}\), as $$ g f r_{t}=\alpha_{0}+\theta_{0} p e_{t}+\delta_{1}\left(p e_{t-1}-p e_{t}\right)+\delta_{2}\left(p e_{t-2}-p e_{t}\right)+u $$ where we omit the other explanatory variables for simplicity. As always with multiple regression analysis, \(\theta_{0}\) should have a ceteris paribus interpretation. Namely, if \(p e_{t}\) increases by one (dollar) holding \(\left(p e_{t-1}-p e_{t}\right)\) and \(\left(p e_{t-2}-p e_{t}\right)\) fixed, \(g f r_{t}\) should change by \(\theta_{0}\) i. If \(\left(p e_{t-1}-p e_{t}\right)\) and \(\left(p e_{t-2}-p e_{t}\right)\) are held fixed but \(p e_{t}\) is increasing, what must be true about changes in \(p e_{t-1}\) and \(p e_{t-2} ?\) ii. How does your answer in part (i) help you to interpret \(\theta_{0}\) in the above equation as the LRP?

Short answer

If \(pe_{t}\) increases, \(pe_{t-1}\) and \(pe_{t-2}\) must increase by the same amount. \(\theta_{0}\) reflects the LRP, showing \(gf r_{t}\) changes by \(\theta_{0}\) for a unit increase in \(pe_{t}\), holding other variables constant.

Step-by-step solution

Analyzing the given equation

The given model is \(gf r_{t}=\alpha_{0}+\theta_{0} pe_{t}+\delta_{1}(pe_{t-1}-pe_{t})+\delta_{2}(pe_{t-2}-pe_{t})+u\). Our goal is to analyze the impact of \(pe_{t}\), \(pe_{t-1}\), and \(pe_{t-2}\) on \(gf r_{t}\).

Setting conditions for constants

To hold \((pe_{t-1} - pe_{t})\) constant when \(pe_{t}\) increases, any increase in \(pe_{t}\) must be matched by a similar increase in \(pe_{t-1}\), ensuring \(pe_{t-1} = pe_{t} + c\), where \(c\) is a constant.

Maintaining the next constant

To keep \((pe_{t-2} - pe_{t})\) constant with \(pe_{t}\) rising, \(pe_{t-2}\) should also increase by an equivalent amount to \(pe_{t}\), resulting in \(pe_{t-2} = pe_{t} + d\), where \(d\) is a constant.

Understanding changes in terms

With \((pe_{t-1} - pe_{t})\) and \((pe_{t-2} - pe_{t})\) constant, \(\delta_{1}(pe_{t-1} - pe_{t})\) and \(\delta_{2}(pe_{t-2} - pe_{t})\) become zero; hence, no change in \(gf r_{t}\) is attributed to these terms when \(pe_{t}\) increments.

Interpreting \(\theta_{0}\)

With the influence of \(\delta_{1}\) and \(\delta_{2}\) neutralized, an additional dollar in \(pe_{t}\) only impacts \(gf r_{t}\) through \(\theta_{0} * 1\). Thus, \(\theta_{0}\) represents the long-run propensity (LRP), signifying the change in \(gf r_{t}\) for a one-dollar change in \(pe_{t}\), ceteris paribus.

Key concepts

Long-Run Propensity

Long-run propensity (LRP) is a key concept in econometrics and multiple regression analysis, referring to the expected change in the dependent variable following a long-term change in one of the independent variables, assuming all else remains constant. In our exercise, the primary focus is on how changes in past values of a predictor impact the dependent variable in the equation given.

When analyzing the model\[gfr_{t} = \alpha_{0} + \theta_{0} pe_{t} + \delta_{1}(pe_{t-1} - pe_{t}) + \delta_{2}(pe_{t-2} - pe_{t}) + u\], one wants to understand the role of \(\theta_{0}\), which represents the long-run propensity of the dependent variable, \(gfr_{t}\), in response to a change in \(pe_{t}\). If \(pe_{t}\) increases by one dollar, keeping \((pe_{t-1} - pe_{t})\) and \((pe_{t-2} - pe_{t})\) constant, \(gfr_{t}\) is expected to change by \(\theta_{0}\).

• LRP provides a ceteris paribus interpretation, meaning it isolates the effect of \(pe_{t}\) on \(gfr_{t}\) while holding other variables fixed.
• This interpretation is essential for understanding long-term economic relationships in regression models.

Multiple Regression Analysis

Multiple regression analysis is a statistical technique used to study the relationships between one dependent variable and multiple independent variables. The primary goal is to explore and quantify how changes in the independent variables influence the dependent variable.

In the context of our exercise, the model is:\[gfr_{t} = \alpha_{0} + \theta_{0} pe_{t} + \delta_{1}(pe_{t-1} - pe_{t}) + \delta_{2}(pe_{t-2} - pe_{t}) + u\]This model explores the impact of the current and past values of \(pe\) on \(gfr\).

• Each coefficient in the model represents the change in the dependent variable for a one-unit change in the corresponding independent variable while holding other variables constant.
• The coefficients for the lagged differences, \(\delta_{1}\) and \(\delta_{2}\), reveal how the past differences between \(pe\) terms affect the dependent variable.
• By taking into account the multiple regressors, we can isolate the unique effect of \(pe_{t}\) on \(gfr_{t}\) through \(\theta_{0}\).

Ceteris Paribus Interpretation

The ceteris paribus interpretation is an essential part of understanding results in regression analysis. "Ceteris paribus" is a Latin phrase that means "all other things being equal." In econometric analysis, this term is used to describe the effect of one variable on the dependent variable while keeping other variables unchanged.

In the regression model discussed, maintaining ceteris paribus allows analysts to interpret \(\theta_{0}\) as the change in \(gfr_{t}\) attributed exclusively to a unit change in \(pe_{t}\), without the influence of other variables.

• To achieve a true ceteris paribus state, changes in \(pe\) must be compensated by equivalent changes in its lagged values \(pe_{t-1}\) and \(pe_{t-2}\), ensuring that \((pe_{t-1} - pe_{t})\) and \((pe_{t-2} - pe_{t})\) remain constant.
• This allows the estimate of \(\theta_{0}\) to be a pure measure of the long-run effect on \(gfr_{t}\).
• The rigorous application of the ceteris paribus condition helps in drawing accurate and meaningful economic conclusions from the model.