Question

Suppose that \(y_{t}\) follows the model $$ \begin{aligned} y_{t} &=\alpha+\delta_{1} z_{t-1}+u_{t} \\ u_{t} &=\rho u_{t-1}+e_{t} \\ \mathrm{E}\left(e_{t} | I_{t-1}\right) &=0 \end{aligned} $$ where \(I_{t-1}\) contains \(y\) and \(z\) dated at \(t-1\) and earlier. i. Show that \(\mathrm{E}\left(y_{t+1} | I_{t}\right)=(1-\rho) \alpha+\rho y_{t}+\delta_{1} z_{t}-\rho \delta_{1} z_{t-1}\). (Hint: Write \(u_{t-1}=y_{t-1}-\alpha-\delta_{1} z_{t-2}\) and plug this into the second equation; then, plug the result into the first equation and take the conditional expectation. ii. Suppose that you use \(n\) observations to estimate \(\alpha, \delta_{1},\) and \(\rho .\) Write the equation for forecasting \(y_{n+1}\) iii. Explain why the model with one lag of \(z\) and \(\mathrm{AR}(1)\) serial correlation is a special case of the model $$y_{t}=\alpha_{0}+\rho y_{t-1}+\gamma_{1} z_{t-1}+\gamma_{2} z_{t-2}+e_{t}$$ iv. What does part (iii) suggest about using models with \(\mathrm{AR}(1)\) serial correlation for forecasting?

Short answer

The model shows AR(1) effectively captures essential dynamics.

Step-by-step solution

Rewriting Previous Period Error

Let's start by expressing the error from the previous period: \( u_{t-1} = y_{t-1} - \alpha - \delta_{1} z_{t-2} \). This expression relates the observed value at time \( t-1 \) minus the influence of \( z_{t-2} \) and the intercept \( \alpha \).

Substitute Previous Error in AR(1) Model

Substitute \( u_{t-1} \) from the previous step into the AR(1) error model: \( u_{t} = \rho (y_{t-1} - \alpha - \delta_{1} z_{t-2}) + e_{t} \).

Substitute AR(1) Error into Original Equation

Plug the expression for \( u_{t} \) into the original model equation: \( y_{t} = \alpha + \delta_{1} z_{t-1} + \rho (y_{t-1} - \alpha - \delta_{1} z_{t-2}) + e_{t} \). Simplifying gives \( y_{t} = (1-\rho) \alpha + \rho y_{t-1} + \delta_{1} z_{t-1} - \rho \delta_{1} z_{t-2} + e_{t} \).

Taking Conditional Expectation

The conditional expectation, given the information set \( I_{t} \), leads to: \( \mathrm{E}(y_{t+1} \mid I_{t}) = (1-\rho) \alpha + \rho y_{t} + \delta_{1} z_{t} - \rho \delta_{1} z_{t-1} \) because \( \mathrm{E}(e_{t+1} | I_{t}) = 0 \).

Forecasting with Estimated Parameters

For forecasting \( y_{n+1} \), plug in the estimated parameters into the conditional expectation model: \( \hat{y}_{n+1} = (1-\hat{\rho}) \hat{\alpha} + \hat{\rho} y_{n} + \hat{\delta}_{1} z_{n} - \hat{\rho} \hat{\delta}_{1} z_{n-1} \).

Model as Special Case of Extended AR Model

Part (iii) aims to understand the connection: our original model becomes a special case because if \( \gamma_{1} = \delta_{1} \), \( \gamma_{2} = -\rho \delta_{1} \), and \( \rho \) from the extended model is \( \rho \) from the original, then the models represent the same dynamics. The condition \( e_t = e_t \) holds, indicating both forms theoretically model the same system.

Implications for Forecasting

IV) The use of \( \mathrm{AR}(1) \) serial correlation in models suggests that even seemingly different models can capture similar properties of time series data. The inclusion of \( z \) lags helps to capture dynamic effects, but the AR(1) structure can streamline complexity without losing essential forecasting power.

Key concepts

Time Series Analysis

Time series analysis involves examining a sequence of data points collected over time. It helps us understand patterns and relationships in data that change over time. For example, we might look at annual temperature changes or daily stock prices. In this approach, each observation in the series depends not only on external factors but often on its own previous values too. By analyzing time series data, you gain insights into what might happen next. Another key point is decomposing the data into trend, seasonality, and random noise components.
This analysis is crucial in econometrics, where understanding past trends can aid in predicting future economic variables, enhancing decision-making processes in economics, finance, and many other fields.

• Patterns can be deterministic, like a consistent uptrend, or stochastic, which may seem random but follow a probability distribution.
• Time series analysis often uses models like AR, MA, or ARIMA to forecast future data points.

Forecasting Models

Forecasting models are mathematical tools designed to predict future values based on past and present data. These models analyze historical data to find patterns that can project future trends. Forecasting is crucial in many disciplines, such as economics, finance, supply chain management, and meteorology. Each field uses specific forecasting models tailored to their unique data characteristics and prediction needs.
To make forecasts, models like ARMA or ARIMA might be employed. These use autoregressive and moving average components to capture relationships between observations and their inertia over time. Using these models, economists and analysts can estimate future trends in unemployment, GDP, stock prices, etc.

• ARIMA stands for Autoregressive Integrated Moving Average, a common forecasting approach for time series data.
• This method is effective for short to medium-term forecasting.

Serial Correlation

Serial correlation, also known as autocorrelation, refers to the relationship between a given variable and a lagged version of itself over successive intervals. In time series data, this often occurs because past values influence future values, violating the assumption that observations are independent.
High serial correlation can significantly affect the accuracy of econometric models. It indicates that the errors from each prediction are not random but follow a pattern. Therefore, it's crucial to test for and account for serial correlation when building a forecasting model. If serial correlation exists, using models that inherently recognize this, such as autoregressive models, can be beneficial.

• Positive serial correlation means that high values are followed by high values, and low values by low values.
• Negative serial correlation means that high values are followed by low values and vice versa.

Autoregressive Models

Autoregressive models are a type of forecasting model used in time series analysis where the variable of interest is regressed on its own previous values. These models help predict future data points by expressing the current variable as a function of its past observations. An AR(1) model, for instance, only relies on the immediately preceding data to make forecasts.
Autoregressive models are particularly useful when the primary predictors of future values are their prior observations. They simplify the forecasting process by focusing on past behavior to forecast future values. The AR part of models like ARMA or ARIMA embodies this predictive feature. Notably, autoregressive models are flexible and can capture different dynamics by adjusting the number of lagged terms included.

• AR models are often denoted as AR(p), where 'p' indicates the number of previous data points considered.
• They are foundational in more complex models, revealing patterns that simpler models might overlook.