Question

Let \(g M_{t}\) be the annual growth in the money supply and let unem, be the unemployment rate. Assuming that unem_ follows a stable AR(1) process, explain in detail how you would test whether \(g M\) Granger causes unem.

Short answer

Test if past values of money supply growth forecast unemployment using an F-test on an extended AR(1) model.

Step-by-step solution

Understanding Granger Causality

Granger causality is a statistical hypothesis test to determine if one time series can forecast another. In this case, we want to see if past values of the growth in money supply (\(g M_t\)) help in predicting the unemployment rate (unem).

Specify the AR(1) Model for Unemployment

Since unem follows a stable AR(1) process, the model is specified as:\[unem_t = \alpha + \beta \cdot unem_{t-1} + \epsilon_t\]where \(\alpha\) is a constant, \(\beta\) is the autoregressive coefficient, and \(\epsilon_t\) is the error term.

Extend the Model to Include g M_t

To test if \(g M_t\) Granger causes unem, extend the AR(1) model by including past values of \(g M_t\):\[unem_t = \alpha + \beta \cdot unem_{t-1} + \sum_{i=1}^{p} \gamma_i \cdot g M_{t-i} + \epsilon_t\]Here, \(p\) is the number of lags for \(g M_t\) and \(\gamma_i\) are the coefficients that determine the influence of past \(g M_t\) on unem.

Conduct Hypothesis Testing

Test the joint significance of the coefficients \(\gamma_i\) by forming the null hypothesis \(H_0: \gamma_1 = \gamma_2 = ... = \gamma_p = 0\). This can be done using an F-test in a regression setup to determine if past values of \(g M_t\) significantly improve the forecasting of unem.

Interpret the Results

If the null hypothesis is rejected, it indicates that the past values of \(g M_t\) do have predictive power for unem, meaning \(g M\) Granger causes unem. If not rejected, there is no evidence to claim Granger causality from \(g M_t\) to unem.

Key concepts

Time Series Analysis

Time series analysis is a statistical technique that deals with time-ordered data points. This type of analysis is essential when working with data that are collected over time, like monthly sales data, daily temperatures, or in this case, annual growth in the money supply and unemployment rates.
Time series analysis aims to extract meaningful statistics and characteristics of the data. Common methods include decomposition of the series, identifying trends, detecting seasonality, and more.
The goal is to use these insights for

• modeling the underlying structures,
• making predictions, and
• understanding how different variables interact with each other over time.

Analyzing time series data is crucial since it can reveal patterns that are not obvious when merely looking at summary statistics or scattered data points.

Autoregressive Model

An autoregressive (AR) model is a type of statistical model used for time series data. It is based on the idea that the current value of the series is related to its past values. This type of model is particularly useful for capturing patterns and making predictions from time-dependent data.
In an AR model, the future value of a variable is assumed to be a linear function of several past observations. For instance, an AR(1) model predicts the current value based on the immediately preceding value. This can be mathematically represented as:\[unem_t = \alpha + \beta \cdot unem_{t-1} + \epsilon_t\] where:

• \(\alpha\) is a constant term,
• \(\beta\) represents the relationship with the past value, and
• \(\epsilon_t\) is the error term accounting for randomness.

This model is called 'autoregressive' because it regresses against itself. It's a fundamental concept in time series analysis, helping analysts to forecast future values based on a simple linear relationship.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or infer conclusions about populations based on sample data. In the context of time series analysis, it helps determine if certain patterns or relationships observed in the sample data can be generalized to the larger population.
The process begins with forming two hypotheses: the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\). The null hypothesis usually states that there is no effect or no relationship between variables being studied.
To test these hypotheses, statisticians use test statistics derived from the data, assessing whether observed results are compatible with the stated null hypothesis. A common threshold in hypothesis testing is the significance level, often set at 5%, which delineates the probability of rejecting the null hypothesis when it is actually true.
If the test statistic meets criteria suggesting low probability that the null hypothesis is true, we reject \(H_0\) and infer support for the alternative hypothesis.

Forecasting

Forecasting involves predicting future values based on past and present information, and it is a vital component of time series analysis. For instance, using an autoregressive model, one may attempt to forecast future unemployment rates by utilizing past unemployment data points.
By leveraging statistical models and past trends, analysts aim to anticipate changes in key metrics, enabling businesses and policymakers to make informed decisions. Forecasting can be achieved through multiple methods, such as:

• Exponential Smoothing
• Moving Averages
• Autoregressive Integrated Moving Average (ARIMA)

The accuracy of forecasts heavily depends on how well the model captures the underlying patterns and dynamics of the time series data. Therefore, selecting the right model and adequately validating it are paramount steps in the forecasting process.

Statistical Hypothesis Test

A statistical hypothesis test is a formal technique that compares two hypotheses using sample data. It is central to methods like Granger causality, which tests if one time series can predict another.
Within this framework, the null hypothesis can be technically expressed. For example, suppose we want to test if the growth in money supply (\(g M_t\)) Granger causes unemployment (unem). Here, the null hypothesis might be formulated as:\(H_0: \gamma_1 = \gamma_2 = \ldots = \gamma_p = 0\)This states that past values of \(g M_t\) do not help predict future unemployment rates (unem). Employing statistical tests like the F-test can determine whether these \(\gamma\) coefficients are jointly significant.
If the calculated value of the test statistic is beyond a critical threshold, we reject \(H_0\), providing evidence that \(g M_t\) indeed has predictive power over unem in the context of Granger causality.