Example

Example

Example

Correlation of a variable with its lag is first-order autocorrelation.

Second-order autocorrelation

Correlation of a variable with its 2nd lag is second-order autocorrelation.

Autocorrelation function

• The autocorrelation function (acf) shows autocorrelation at multiple lags.
• Lag 0 is usually presented too, even though the correlation at lag 0 is always 100%.
• The band is a confidence interval under the null hypothesis that the autocorrelation is zero.
• Estimates outside the band are statistically significant.

Fama-French factors

• Ask Julius to use pandas datareader to get the monthly Fama-French factors for the maximum history available.
• Ask Julius to plot the acf’s of Mkt-RF, HML, and SMB.

First-order autoregression

• A first-order autoregression is the equation \[x_t = \alpha + \beta x_{t-1} + \varepsilon_t\]
• Positive \(\beta \Leftrightarrow\) positive first-order autocorrelation.
• Positive beta \(\Rightarrow\) persistence.

Higher order autoregressions

• A \(p\)th order autoregression is the equation \[x_t = \alpha + \beta_1x_{t-1} + \cdots + \beta_px_{t-p} + \varepsilon_t\]
• There are standard methods for choosing the optimal \(p\), trading off goodness of fit and parsimony.
• Ask Julius to fit an AR(p) to HML and find the optimal \(p\).

What should we try to forecast?

• Price of AAPL?
• Change in price of AAPL?
• Percent change in price of AAPL (return)?
• A forecast of changes or percent changes implies a forecast of the price and vice versa, so only need to forecast one of them directly.
• What variable should we use in an autoregression?

Stationarity

• One issue is that autocorrelation or autoregression estimates are reliable only for stationary variables.
• Prices grow over time (unstationary).
• Changes in prices become larger over time (in absolute value)
• Returns are the right thing to use in an autoregression.

Other examples

• Autoregression for crude oil price or change in crude price or percent change in crude price?
• Autoregression for interest rate or change in interest rate or percent change in interest rate?

Autoregression and \(\Delta x\)

Rearrange \[x_t = \alpha + \beta x_{t-1} + \varepsilon_t\] as \[\Delta x_t = \alpha + (\beta-1) x_{t-1} + \varepsilon_t\] \[\Delta x_t = (\beta-1)\left(x_{t-1} - \frac{\alpha}{1-\beta}\right) + \varepsilon_t\]

Mean reversion

• An AR(1) is \[\Delta x_t = (\beta-1)\left(x_{t-1} - \frac{\alpha}{1-\beta}\right) + \varepsilon_t\]

• \(\beta<1\) implies regression towards the mean.

• The mean is \(\alpha/(1-\beta)\).

• \(|\beta-1|\) is called the rate of mean reversion.

• \(\beta>1\) implies nonstationary.

Example

Ask Julius to simulate the process x_t = 1 + 0.5*x_{t-1} + e_t by drawing 1,000 standard normals for e_t starting at x_0=1. Ask Julius to plot the process.

Forecasts

• Given today’s value \(x_t\), the AR(1) forecast for the next period’s value is \[\hat{x}_{t+1} = \alpha + \beta x_t\]
• The forecast for the period after that is \[\hat{x}_{t+2} = \alpha + \beta \hat{x}_{t+1}\]
• Etc.

Forecast Convergence

• The forecasts will converge to \(\alpha / (1-\beta)\) as we look further out into the future.
  
• They will converge quickly is \(\beta\) is small. \[ \beta \;\; \text{small} \quad \Rightarrow \quad 1-\beta \;\; \text{large}\]

Interest rates

• Ask Julius to use pandas-datareader to download 10-year Treasury yields starting in 1980 from FRED.
• Ask Julius to fit an AR(1) and to use it to forecast the yield for the next 20 days.

Vector Autoregression

• Multiple variables (vector)
• One equation for each variable. Lags of all variables are used as predictors, including the variable’s own lag.
• Can answer questions like: does this month’s SMB return forecast next month’s HML return?

Example

• Ask Julius to use pandas datareader to download the monthly 5 Fama-French factors since 1970 from French’s data library.
• Ask Julius to fit a VAR(1) to MKT-RF, SMB, HML, RMW, and CMA and to print the summary to a text file.