Question

Suppose \(y_{t}\) follows a second order FDL model: $$y_{t}=\alpha_{0}+\delta_{0} z_{t}+\delta_{1} z_{t-1}+\delta_{2} z_{t-2}+u_{t}$$ Let \(z^{*}\) denote the equilibrium value of \(z_{t}\) and let \(y^{*}\) be the equilibrium value of \(y_{t},\) such that $$y^{*}=\alpha_{0}+\delta_{0} z^{*}+\delta_{1} z^{*}+\delta_{2} z^{*}$$ Show that the change in \(y^{*},\) due to a change in \(z^{*},\) equals the long- run propensity times the change in \(z^{*}\) $$\Delta y^{*}=L R P \cdot \Delta z^{*}$$ This gives an alternative way of interpreting the LRP.

Short answer

The change in equilibrium \( y^* \) due to change in \( z^* \) is equal to LRP times the change in \( z^* \).

Step-by-step solution

Understanding the Model

The given model is a second-order Finite Distributed Lag (FDL) model. This type of model explains each observation of the dependent variable, here \( y_t \), as a function of the current and past values of another variable, \( z_t \). The equation capturing this relationship is given by: \[ y_{t} = \alpha_{0} + \delta_{0} z_{t} + \delta_{1} z_{t-1} + \delta_{2} z_{t-2} + u_{t} \] where \( \alpha_0 \) is a constant and \( u_t \) is the error term.

Find Equilibrium Values

In equilibrium, the values of variables do not change over time. Thus, for equilibrium values \( y^* \) and \( z^* \), we can express: \[ y^{*} = \alpha_{0} + \delta_{0} z^{*} + \delta_{1} z^{*} + \delta_{2} z^{*} \] This simplifies to: \[ y^{*} = \alpha_{0} + (\delta_{0} + \delta_{1} + \delta_{2}) z^{*} \]

Define Long-Run Propensity (LRP)

The Long-Run Propensity (LRP) is the sum of coefficients of \( z_t \), \( z_{t-1} \), and \( z_{t-2} \). It captures the total change in \( y \) given a sustained change in \( z \). Mathematically, it is defined as: \[ LRP = \delta_{0} + \delta_{1} + \delta_{2} \]

Compute Change in Equilibrium Values

We now express the change in equilibrium values, \( \Delta y^* \), due to a change in \( \Delta z^* \). From Step 2, we've:\[ \Delta y^{*} = (\delta_{0} + \delta_{1} + \delta_{2}) \Delta z^{*} \]This equation translates to:\[ \Delta y^{*} = LRP \cdot \Delta z^{*} \]Thus, the change in \( y^* \) due to a change in \( z^* \) is equivalent to the Long-Run Propensity multiplied by the change in \( z^* \).

Key concepts

Equilibrium Value

The equilibrium value in the context of a finite distributed lag model refers to a state where the variables involved do not change over time. In such a state, everything is balanced and constant. For the model given, the equations find their equilibrium values for both dependent and independent variables, represented as \( y^* \) and \( z^* \).

The equilibrium value of \( y^* \) is determined by holding \( z_t \), \( z_{t-1} \), and \( z_{t-2} \) all constant at a value \( z^* \). This means that the dependent variable \( y_t \) reaches a steady state where it no longer fluctuates. This is mathematically expressed as:

\[ y^{*} = \alpha_{0} + \delta_{0} z^{*} + \delta_{1} z^{*} + \delta_{2} z^{*} \]

In terms of practical understanding, when discussing equilibrium, it means looking at the long term point at which the effects of previous changes and interventions have stabilized. In essence, understanding equilibrium helps in predicting future values given no further shocks to the system.

Long-Run Propensity

Long-Run Propensity (LRP) is a key concept that describes the overall effect of a sustained change in the independent variable on the dependent variable over time. In this finite distributed lag model, LRP captures how much the equilibrium value of \( y^* \) changes if \( z^* \) experiences a consistent shift.

To calculate LRP, we need to add up the effects of present and past values of \( z \) on \( y \). Specifically, it is the sum of the coefficients \( \delta_0 \), \( \delta_1 \), and \( \delta_2 \). This can be represented as:

\[ LRP = \delta_{0} + \delta_{1} + \delta_{2} \]

The LRP tells us that an increase in \( z^* \) by one unit results in an increase in \( y^* \) by exactly the amount of \( LRP \). This is because the LRP effectively sums the impacts of all lag periods, giving a holistic view of a change's total effect. Through this metric, the model helps in understanding the accumulated long-term effects stemming from a change in \( z \).

Change in Variables

In this model, the change in variables sheds light on the dynamic relationship between \( y^* \) and \( z^* \). When \( z^* \) changes, the equilibrium value of \( y^* \) also changes by an amount dictated by the long-run propensity (LRP). This change can be quantified and is vital for understanding how adjustments in one variable propagate through the system.

Mathematically, if there is a shift in \( z^* \), the corresponding shift in \( y^* \) can be expressed as:

\[ \Delta y^{*} = LRP \cdot \Delta z^{*} \]

This equation is crucial, as it shows how the input variable's changes translate into changes in the output variable within the equilibrium framework. This relationship highlights the proportional adaptation of \( y^* \) corresponding to any modification in \( z^* \). For example, if \( z^* \) increases by 2 units, \( y^* \) increases by \( 2 \times LRP \) units.

Such insights help in predicting and managing the interplay between factors in various real-world scenarios, where understanding the ripple effects of changes is essential for decision-making and forecasting.