6

Univariate Time Series Modelling and Forecasting

• This chapter discusses predicting future values based solely on past data.
• Past values contain important information for forecasting.

Some Notation and Concepts

• Strictly Stationary Process: The probability structure of a sequence remains the same over time. The probability of future values, given previous values, is unchanged for any time shift.
• Weakly Stationary Process: Characterized by: constant mean (over time), constant variance (over time), and autocovariances that only depend on the length (not the starting point) of the time lag between observations.
• Autocovariances ("γ") are covariances between observations at different time periods.
• Autocorrelations ("τ") are standardized autocovariances, making them unitless and simpler for analysis.
• Autocorrelation functions (ACF) are the plot of τ against different time lags (s).

A White Noise Process

• A white noise process has no discernible structure, and its autocorrelations are essentially zero, except at lag zero (τ = 1 for s=0).
• The ACF is approximately normally distributed with mean 0 and variance 1/T, where T is the sample size.
• White noise allows testing autocorrelations’ statistical significance.

Joint Hypothesis Tests

• Statistical test for multiple autocorrelations simultaneously equal to zero.
• Q-statistic and Ljung-Box statistic can be used for this test.

An ACF Example

• Example illustrates testing individual and joint significance of autocorrelation coefficients. A practical application of determining if lagged values can predict future values of a series.

Moving Average Processes

• A qth order moving average model (MA(q)) expresses values as a linear combination of the current and previous q random errors (u).
• The Mean and variances are based on the error term random variable characteristics.

Example of an MA Problem

• Examples and exercises help students apply MA concept to a real-world financial scenario. This helps clarify the role of understanding how moving average models predict variables and their relationships.

Solution to MA Problem

• Specific example analysis, calculation of the mean and variance of X, calculation of ACF.

Autoregressive Processes

• An AR(p) model represents a series as a linear combination of previous values and a current error term.
• A key concept in time series analysis is the lag operator. This idea gives a concise way to describe models for understanding and calculating related statistical measures.

The Stationary Condition for an AR Model

• A stationarity condition for AR models needs all roots (from the AR process characteristic equation characteristic equation) to be outside the unit circle. This is to ensure stability and a defined behavior over time.

Wold's Decomposition Theorem

• Any stationary time series can be broken down into a purely deterministic component and a purely stochastic component (an infinite moving average).

The Moments of an Autoregressive Process

• Describes how to obtain mean, autocovariances, and autocorrelations for AR processes. The Yule Walker equations are key to this process.

Sample AR Problem

• Example demonstrating calculation of mean, variance, and autocorrelation function for a simple AR(1) model. This further refines understanding time-lagged values and how they relate to one another and predict future values.

Univariate Time Series Forecasting in Economics

• Forecasting is the process of predicting future values of a time series.
• Different methods are used to forecast.

In-Sample Versus Out-of-Sample

• In-sample means modeling with the entire dataset and evaluating accuracy on this same dataset.
• Out-of-sample is modeling with part of the dataset and testing on the remaining part. This process is crucial validation.

How to Produce Forecasts

• Different forecasting methods are presented in order to cover different aspects of such tasks.

Models for Forecasting

• Structural models (e.g., regression) use external factors.
• Time series models use past values of the variable itself.

Forecasting with ARMA Models

• Forecasting equations for ARMA models.

Forecasting with MA Models

• Forecasting for MA models (based on the MA order).

Forecasting with AR Models

• Forecasting for AR models (based on the AR order).

How can we test whether a forecast is accurate or not?

• Model accuracy evaluation and methods.

Forecast Evaluation Example

• Example application of MSE, MAE, and percentage of correct sign predictions.

What factors are likely to lead to a good forecasting model?

• Factors involved in accurate model building.

Statistical Versus Economic or Financial loss functions

• Comparing and contrasting various forecasting metrics. How well the model performs in real-world application.

Back to the original question: why forecast?

• Reasons for forecasting and the problems of judgemental forecasts.

Exponential Smoothing

• Method using previous values and weights to forecast the future.

Exponential Smoothing (cont'd)

• Method using weights based on recent data history for calculating smoothed values from past data.

Exponential Smoothing (cont'd)

• Characteristics, advantages, and disadvantages, limitations in dealing financial data.

Seasonal adjustments in Exponential smoothing

• Methods addressing seasonality in time series data.

ARIMA Models

• Integrated Models: accounting for data integration.