Reading 13: Time-Series Analysis Flashcards
Calculate and evaluation the predicted trend value for a time series, modeled as either a linear trend or a log-linear trend, given the estimated trend coefficients.
Linear trend: substitute the time period you are looking for for the ‘t’ in the equation. Which is multiplied by the coeffiient and added to the intercept
Log-Linear trend: take the ln ‘natural log’ of both sides. Remember to CONVERT BACK to EXPONENTIAL form
Describe factors that determine whether a linear or a log-linear trend should be used with a particular time series, and evaluate the limitations of trend models.
Linear Factors:
Log-Linear Factors:
Limitations of trend models
Explain the requirement for a time series to be covarieance stationary, and describe the significance of a series that is not stationary.
Dealing with AR (autoregressive models):
Covarianec Stationary: Must have CONSTANT and FINITE:
- Expected Value
- Variance
- Covariance with leading or lagged values
Not stationary: will produce meaningless regression results!
Describe the structure of an autoregressive (AR) model of order p, and calculate one- and two-period-ahead forecasts given the estimated coefficients.
Structure of an AR model: the dependent variable is regressed against previous values of itself (think of an excel doc example). The distinction between dependent and independent variables no longer matters as X is the ONLY variable.
Calculate one-period-ahead (AR1): plug in Xo ans solve for X1
Calculate two-period-ahead (AR2): plug in X1 and solve for X2
Explain how autocorrelations of the residuals can be used to test whether the autoregressive model fits the time series.
IF SERIAL CORRELATION EXISTS, then you should ADD more lags to the model. i.e. increase from AR1 to AR2, etc…. UNTIL no serial correlation exists
YOU CANNOT use the Durbin-Watson statistic with AR models. You must use a t-test
(1) test for serial correlation using a t-test
(2) IF serial correlation exists, increase ‘order’ or lag periods until serial correlation no longer exists
Explain mean reversion, and calculate a mean-reverting level:
Mean reversion:the value of the dependent variable tends to fall when above its mean and rise when below its mean
Calculate:
For an AR (1) model: MRL = Bo / (1-b1)
Contrast in-sample and out-of-sample forecasts, and compare the forecasting accuracy of different time-series models based on the root mean squared error of the criterion.
In-sample:
Out-of-sample:
Root mean squared error of the criterion: WANT LOW RMSE
Explain the instability of coefficients of time-series models:
Instability: estimated regression coefficients change from one time period to another.
- Creates a trade-off between long track records which are statistically stable versus shorter time periods where you can more easily match process/environments
Describe characteristics of random walk processes, and contrast them to covariance stationary processes.
Characteristics of random walk processes: is non-stationary process with an undefined mean reverting level
Covariance stationary processes:
Describe implications of unit roots for time-series analysis, explain when unit roots are likely to occur and how to test for them, and demonstate how a time-series with a unit root can be transformed so it can be analyzed with an AR model.
Implications of unit roots: need to be fixed with first differences by creating a new variable by subtracting the data and lagged date from each other.
Explain unit roots are likely to occur: common when a variable consistently increases or decreases over time
Demonstrate a transformation: USE FIRST DIFFERENCES
Describe the steps of the unit root test for nonstationarity, and explain the relation of the test to autoregressive time-series models.
Steps of the unit root test for nonstationarity:
Relation to autoregressive time-series models:
Explain how to test and correct for seasonality in a time-series model, and calculate and interpret a forecasted value using an AR model with a seasonal lag.
Test for seasonality: bring in the seasonal component with proper lag
Correction for seasonality: add a seasonal lag ALONGSIDE the other AR1 lag
Interpret:
Explain autoregressive conditional heteroskedasticity (ARCH), and describe how ARCH models can be applied to predict the variance of a time series.
Explain ARCH:
- The variance of the residuals is not constant
- The variance of the residuals in one time period is correlated to the residuals in another time period
- The SE of the coefficients in the AR models are unreliable
- Generalized least squares (GLS) to correctly estimate the SEs
IF a1 is signicant the time series is ARCH(1)
Describe application to predict variance:
- DO NOT GET BOGGED DOWN
Explain how time-series variables should be analyzed for nonstationary and/or cointegration before use in a linear regression.
How to analyze for nonstationary:
- If they are both covariance statoinary, you’re ok
- If only one of time series are covariance stationary, there is nothing we can do
- IF Neither time series is covariance stationary….. Check for cointegration
How to analyze for cointegration:
- Defined: two time series are related to the same macro variables or follow the same trend
- If they are co-integrated, go ahead and use the model
Determine an appropriate time-series model to analyze a given investment problem, and justify that choice.
Explain by looking at the flowchart in the material
