# Auto-Regressive Models - Random Walks and Unit Roots

- This is the case of an AR time series model where the predicted value is expected to equal the previous period plus a random error:

xt = b0 + xt-1 + εt

- When b0 is not equal to zero, the model is a random walk with a drift, but the key characteristic is a b1 = 1.
- The expected value of the error is still zero.
- The mean reverting level for a random walk is not covariance stationary and the technique of first differencing is frequently used to transform an AR model with one time lag variable (AR1) into a model that is covariance stationary.
- If an AR time series is covariance stationary, then the serial correlations for the lag variables are insignificant or they rapidly drop to zero as the number of time period lags rises.
- When the lag coefficient is not statistically different from 1, a unit root exists.
- Dickey-Fuller test = applied to AR1 model to test for a unit root.
- If a unit root is present, then the model is not covariance stationary; if this is the case, the independent variable must be transformed, so you can re-model.

LESSONS

- CFA L2: Quantitative Methods - Introduction
- Quants: Correlation Analysis
- Quants: Single Variable Linear Regression Analysis
- Standard Error of the Estimate or SEE
- Confidence Intervals (CI) for Dependent Variable Prediction
- Coefficient of Determination (R-Squared)
- Analysis of Variance or ANOVA
- Multiple Regression Analysis
- Multiple Regression and Coefficient of Determination (R-Squared)
- Fcalc – the Global Test for Regression Significance
- Regression Analysis and Assumption Violations
- Qualitative and Dummy Variables in Regression Modeling
- Time Series Analysis: Simple and Log-linear Trend Models
- Auto-Regressive (AR) Time Series Models
- Auto-Regressive Models - Random Walks and Unit Roots
- ARMA Models and ARCH Testing
- How to Select the Most Appropriate Time Series Model?

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