What is meant by dynamic model? Explain how the following model can be estimated? 𝑦𝑡 =∝ +𝛽𝑥𝑡 + 𝛾𝑦𝑡−1 + 𝑢𝑡 where |𝛾| < 1 and 𝑢𝑡 = 𝜌 𝑢𝑡−1+ 𝜀𝑡

Introduction

In econometrics, a dynamic model is one that includes lagged values of the dependent or independent variables. These models are particularly useful when analyzing time series data where past events influence current outcomes. Dynamic models are essential for studying the adjustment process and persistence over time.

What is a Dynamic Model?

A dynamic model incorporates past or lagged values of the dependent variable (and sometimes the independent variables) as regressors. This allows the model to capture temporal dependencies and behavioral inertia in economic relationships.

General form:

y_t = α + βx_t + γy_t-1 + u_t

Where:

y_t: current value of the dependent variable
x_t: current value of the independent variable
y_t-1: lagged value of dependent variable (dynamic component)
u_t: error term

The Given Model

We are given:

y_t = α + βx_t + γy_t−1 + u_t

Where the error term follows:

u_t = ρu_t−1 + ε_t, with |γ| < 1 and |ρ| < 1

Understanding the Model

This is a dynamic regression model with autocorrelated errors. The presence of y_t−1 introduces dynamics, and the autocorrelation in u_t implies that the model suffers from serial correlation in errors, which violates OLS assumptions.

Challenges in Estimation

Autocorrelation in the error term (u_t) violates OLS assumptions.
Endogeneity may arise due to correlation between y_t−1 and u_t.

Estimation Methods

Several methods can be used to estimate such models:

1. Ordinary Least Squares (OLS)

Not appropriate due to serial correlation in u_t and possible endogeneity of lagged dependent variable.
OLS estimates will be biased and inconsistent.

2. Cochrane-Orcutt Procedure

This method transforms the model to eliminate serial correlation.
But not ideal when lagged dependent variable is included, due to endogeneity.

3. Generalized Least Squares (GLS)

GLS accounts for autocorrelation in the error term.
May provide consistent estimates, but assumptions about the structure of u_t are required.

4. Instrumental Variables (IV) or Two-Stage Least Squares (2SLS)

Used to handle endogeneity of y_t−1.
Lagged values like y_t−2 or x_t−1 can be used as instruments.
Provides consistent estimates even in the presence of endogenous regressors.

5. Generalized Method of Moments (GMM)

Suitable for dynamic panel data models or large sample time series.
Efficient under weaker assumptions.

Steps to Estimate the Model

Test for autocorrelation (e.g., using Durbin-Watson or Breusch-Godfrey test).
If present, use transformation or instrumental variables to handle it.
Use Two-Stage Least Squares (2SLS) or GMM to address endogeneity of y_t−1.
Check stationarity of the series (ADF test), as non-stationary data can lead to spurious results.
Interpret coefficients carefully, as dynamics can complicate long-run effects.

Interpretation

γ: measures the influence of the past value of y on the current y
β: shows the contemporaneous effect of x_t on y_t
ρ: represents the level of autocorrelation in the error term

Conclusion

The given model is a dynamic model with autocorrelated errors. Such models cannot be estimated using standard OLS due to endogeneity and serial correlation. Techniques like instrumental variables, GLS, or GMM are used to ensure consistent and efficient estimation. Understanding the structure of the dynamic model is essential for choosing the right estimation approach and making valid inferences.