Financial Econometrics

• Time is an amazing factor - a little baby can grow up over time, the weather will change over time, stock prices can change over time
• The purpose of time series analysis is to understand the relationships among data values over time
• Ways to present time series data

  • Auto regressive model
  • Moving average model
  • Combination of auto regressive and moving average models
• Auto regressive model


  The data value at the current time spot is built on top of the data values at the previous time spots
• Moving average model


  The current data value is a result from previous unexpected events
• ARMA model


  Combination of auto regressive and moving average models
• The first step in time series analysis is to ensure the data follows a stationary data assumption - constant mean and constant variance throughout the entire time series
• Differencing


  1. Subtract previous values from the current value

  2. Can do one-lag differencing, two-lag differencing, etc.

  3. Used to transform non-stationary data into stationary data
• Differencing techniques can be applied to process seasonal data and transform it into stationary data
• The techniques and models learned in this lecture series can be applied to a broader range of disciplines beyond financial time series data analysis, such as AI, machine learning, and pattern recognition
• Univariate models


  Variables are modelled and predicting using only information contained in: Own past values, Current and past values of an error term
• Univariate models


  • Different from multivariate models where relationships between several variables are studied
  • Usually a-theoretical – models are constructed to capture important features of the data, not based on theory
  • Useful when the factors influencing the variable of interest are not observable or are measured at a lower frequency
• Univariate model example


  • We want to predict daily stock returns but possible explanatory variables like macroeconomic conditions are available quarterly
• ARIMA


  The most common class of univariate time series models
• ARIMA model components


  1. Integration or stationarity (I)

  2. Auto-regressive processes (AR)

  3. Moving Average processes (MA)
• Stationarity


  A process is stationary if: its mean and variance are constant over time, The covariance between two time periods depends only on the gap between the two time periods and not the time at which the covariance is computed
• Weak stationarity


  The mean, variance and autocovariance function of the process do depend on time.

  A time series which violates the weak stationarity requirements is a non-stationary or integrated time series
• Strictly stationary


  All the moments of the probability distribution (not just the mean and variance) are invariant over time
• Autocorrelation Function (ACF)


  • Values of autocovariance vary with unit of measurement, so autocorrelations are usually preferred
  • The graph of the autocorrelations against lags is the correlogram or autocorrelation function and it completely characterizes a time series
• When a process is stationary


  We can make predictions about the future by generalizing the model to future time periods
• When a process is non-stationary


  We cannot generalize the model for several reasons
• Non-stationary processes cannot be used for forecasting
• White noise process


  A purely random process with mean zero, constant variance and is serially uncorrelated
• The error term that we assume under the classical linear regression model is a white noise error term
• Gaussian white noise process


  A white noise process that also follows a normal distribution
• Standard regression models can be used, but nothing much to estimate
• Moving average processes


  The simplest class of time series models - it is a linear combination of white noise processes
• Moving average process


  1. Depends on the current and previous values of a white noise error term

  2. Is a qth order moving average model, denoted MA(q), where θ is a constant
• MA(1) process


  A first order MA process where Xt = θεt-1 + εt
• For an MA(1) process, Xt has the following properties: E(Xt) = 0, Var(Xt) = (1 + θ^2)σ^2, Cov(Xt, Xt-k) = 0 for k ≠ 1, Cov(Xt, Xt-1) = θσ^2
• L


  Lag operator
• Autoregressive (AR) model


  A model where the current value of the variable depends on past values that the variable took plus an error term
• AR model of order p, denoted as AR(p)


  1. Where μ is the constant and ε_t is a white noise error term

  2. The lag operator notation can be used to write: φ(L)y_t = μ + ε_t

  3. Where φ(L) is known as the characteristic equation
• Stationarity of an AR process

  • Stationarity is a desirable property for an AR model
  • If non-stationary, previous values of the error term will have an increasing effect on the current value of y_t - this is empirically unlikely
  • The stationarity of an AR process depends on the coefficients of the model
  • Generally, to verify stationarity, we 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, we simply require |φ| < 1
• Properties of stationary AR(1) process


  • E[y_t] = μ/(1-φ)
  • Var[y_t] = σ^2/(1-φ^2)
  • The autocorrelation functions shows that the AR process has decaying memory
• ACF


  Autocorrelation function of a process
• PACF


  Partial autocorrelation function, measures the correlation between X_t and X_t-k after controlling for observations in between
• The PACF is useful for checking for hidden information in the residuals
• Correlogram


  Plots of the ACF and PACF over different lags
• Correlograms


  • Can be useful for model selection (values of p and q to feed into AR(p) and MA(q) models)