Financial Econometrics: Time Series Analysis

• Financial econometrics focuses on financial applications of econometrics
• Most financial data are time series

Financial Econometrics vs Econometrics

• Financial econometrics emphasizes time series methods due to the prevalence of time-series data in finance.
• Standard and time series econometrics differ:
  • Differences in data arising from high frequency, seasonality, volatility.
  • Less concern about measurement error, focusing on actual trades or quotes.
  • Financial data can be noisy, making it hard to separate trends from random noise.
  • Random sampling often violated due to correlations over time

Applications of Financial Econometrics

• Examination of how Sri Lanka's stock market index correlates with macroeconomic variables like sovereign debt, inflation, and growth.
• Analysis of the impact of Sri Lanka's exchange rate on its trade deficit.
• Measurement and prediction of stock market return volatility.
• Determining the relationship between a company's size and its share investment returns.
• Assessing the influence of earnings or dividend announcements on stock prices.

Time Series Concepts

• Common time series concepts include:
  • Univariate models
  • Stationarity or integrated processes
  • White noise process
  • Moving average process
  • Autoregressive process
  • Autocorrelation function and partial autocorrelation function

Univariate Models

• Model specifications predict variables using only information:
  • Own past values
  • Current and past values of an error term
• These models differ from multivariate models, which study relationships between multiple variables.
• Univariate models are a-theoretical, designed to capture data features without relying on theory.
• These models are valuable when key factors affecting a variable are unobservable or measured at a lower frequency.
  • An example would be predicting daily stock returns with quarterly macroeconomic data.

ARIMA Model

• ARIMA (Autoregressive Integrated Moving Average) models are the most common class of time series models.
• ARIMA components include:
  • Integration or stationarity (I)
  • Auto-regressive processes (AR)
  • Moving Average processes (MA)
• Understanding the properties of these components allows constructing appropriate models for a dataset.

Stationarity

• A time series is stationary if:
  • Its mean and variance are constant over time
  • Covariance between two time periods depends only on the gap between those periods, not the time at which the covariance is computed
• A weakly stationary or covariance stationary process has constant mean and variance, with time-invariant covariance.
• A non-stationary or integrated time series violates weak stationarity requirements.
• Strict stationarity requires all moments of the probability distribution to be invariant over time, not just the mean and variance

Weak Stationarity

• It is also called covariance stationarity
• Requires all the following conditions to be true:
  • Constant mean
  • Constant variance
  • Autocovariance depends on the time lag (interval between observations) and not on the actual time the variable is observed

Autocorrelation Function (ACF)

• The values of autocovariance depend on the unit of measurement, so autocorrelations are preferred.
• For a stationary process, the graph of autocorrelation ( ) against lags is known as the correlogram or autocorrelation function.
• This function fully characterizes a time series.

Importance of Stationarity

• For stationary processes, future predictions may be made by extending an existing model.
• Non-stationary processes cannot be generalized because of several reasons:
  • Non-stationary processes cannot be used for forecasting
  • Methods of detecting and working with non-stationarity will be detailed later.

White Noise Process

• The white noise process is purely random:
  • The mean is zero
  • Variance is constant
  • Serially uncorrelated
• Under the classical linear regression model, the error term is a white noise error term
• A Gaussian white noise process has a normal distribution
• Standard regression models may be used, but there is nothing to estimate

Moving Average Processes

• They form the simplest class of time series models, a linear combination of white noise processes.
• A moving average process depends on current and previous values of white noise error term.
• The qth order moving average model is denoted MA(q), where is a constant.
• A first order MA process or MA(1) process would simply be

Some Notation

• Lag operator notation is used.
• MA(q) process can be written as:
• Here, ignore the constant

MA(1) Process

• For the MA(1) process:
  • The mean is 0
  • The variance is

Autoregressive Processes

• An autoregressive (AR) model calculates the current value of a variable based on its past values plus an error term.
• An AR model of order p is denoted as AR(p).
• Represents the constant and represents the white noise error term.
• Using the lag operator notation:
• The characteristic equation is

Stationarity of an AR Process

• Stationarity is a desired property for an AR model.
  • If the model is non-stationary, previous values of the error term will have an increasing effect on the current value, which is empirically unlikely.
• Stationarity is dependent on the coefficients of the model.
• To verify stationarity, consider the roots of the characteristic equation.
• If all the roots of this equation are greater than 1, the process is stationary.
• For an AR(1) process, simply require

Example – Random Walk

• Model:
• Rewriting the model using lag notation
• The characteristic equation:
• The root of this equation is not greater than 1, the process is not stationary.
• This process is also called a random walk.

Properties of Stationary AR(1) Process

• The AR(1) process:
  • Has a mean of 0
  • Has a variance
• exhibits the following properties:
• Autocorrelation functions show that the AR process has decaying memory.

ACF and PACF

• The autocorrelation function (ACF) describes the correlation of a process.
• The partial autocorrelation function (PACF) measures correlation between observations, controlling for intermediate observations.
• It is useful for checking for hidden information in the residuals
• A correlogram represents plots of ACF and PACF over different lags, and is used for model selection to determine AR(p) and MA(q) model values.

Sample ACF and PACF

• For an AR(p) model, the typical ACF decays exponentially or has a damped sine wave pattern, whereas the PACF shows significant spikes through lag p.
• For a MA(q) model, the typical ACF shows significant spikes through lag q, whereas the PACF declines exponentially.