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MindMap Gallery CFA Level 2 Time Series

CFA Level 2 Time Series

Time series trend model, autoregressive model*AR time series forecasting steps Covariance stationarity, etc.

Edited at 2019-12-30 14:43:49

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CFA Level 2 Time Series

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Outline

CFA-level one quantitative analysis mind map
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PlotWizard
Quantitative Analysis(1)
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Ava Harper

Time Series

Trend Models

linear trend

A fixed amount that grows over time

log-linear trend

have exponential growth (exponential growth)

A fixed rate that grows over time

predicted trend value of yt

growth rate

Linear trend regression will have the problem that the regression error is related to the observed value. The log will be corrected some, but it is not solved.

Testing for Correlated Errors

DW-test

H0: There are no sequence-related problems,

The premise of using the trend model is that the covariance is stationary. If the covariance is not stationary, the model will be invalid.

Autoregressive (AR) Time-Series Models

We must assume that the time series we are modeling is covariance stationary

Covariance-Stationary Series

the expected value of the time series must be constant and finite in all periods

the covariance of the time series with itself for a fixed number of periods in the past or future must be constant and finite in all periods

the variance of the time series must be constant and finite in all periods

How to check whether the covariance is stationary? Look directly at the plot, if the plot shows the same mean and variance

Autocorrelation coefficient for all lag values = 0

A random walk

Previous value Unpredictable random term

previous period plus an unpredictable random error

incovariance stationary

If the time series is a random walk, it is not covariance stationary

Random walk with drift

A random walk with drift is a random walk with a nonzero intercept term.

Has a unit root

All random walks have unit roots.

If a time series has unit roots, it is impossible to have stationary covariances

Treatment of unit roots

first-differencing the time series; (first-order splitting), perform autoregressive estimation of the sequence after first-order splitting.

Moving-Average Time-Series Models

moving average

Lags behind actual data and plays a role in smoothing data (such as smoothing seasonal fluctuations)

Because of the lag, the prediction effect cannot be achieved.

MA(1) model

MA(q)：A qth order moving-average model

ts first q autocorrelations are nonzero while autocorrelations beyond the first q are zero.

ARMA models

autoregressive moving average models

the parameters in ARMA models can be very unstable;

determining the AR and MA order of the model can be difficult;

ARMA models may not forecast well

ARCH

Autoregressive conditional heteroskedasticity modelAutoregressive conditional heteroskedasticity model

If the coefficient on the squared residual is statistically significant, the time-series model has ARCH(1) errors

if a time-series model has ARCH(1) errors

Multivariate time series problem

All timelines have no cell roots and regression is available

If neither of the time series has a unit root, then we can safely use linear regression.

Only one of the time series has a unit root, can regression be used?

If one of the two time series has a unit root, then we should not use linear regression

All series have unit roots, and time series are cointegrated, regression is available

If both time series have a unit root and the time series are cointegrated, we may safely use linear regression

All series have unit roots, and the time series is not cointegrated, so regression is not available.

however, if they are not cointegrated, we should not use linear regression

(Engle–Granger) Dickey–Fuller test cointegration test

The (Engle–Granger) Dickey–Fuller test can be used to determine if time series are cointegrated

Some issues with time series

Covariance stationarity will form mean reverting

Compare the accuracy of different regression models

The root mean squared error (RMSE) root mean squared error: Error squared and square root of mean

The smaller the better

The parameters of the time series model will be unstable. When using the time series model for estimation, it is necessary to check whether the time series is stable.

The steps of time series forecasting

Understand your investment problem and choose an initial time series model

regression model

Use one variable to predict another variable

time-series model

Predict the same variable using previous data on the same variable

If you use a time series model, first draw a graph to see if the covariance is stationary.

Does not contain

a linear trend a linear trend

an exponential trend an exponential trend

seasonalality

There is a significant deviation in the data within the sample interval, a significant shift in the mean or covariance.

step

Draw a graph to check whether a linear trend or an exponential trend makes the most sense

Estimate trend parameters

Calculate remaining residuals

Durbin–Watson statistic detection sequence related issues

if it does not exist

Model available

if exists

Use autoregressive model autoregressive model

autoregressive model

Treatment of covariance stationarity violations of stationarity

a linear trend,

first-difference the time series.

exponential trend

take the natural log of the time series and then first-difference it

shifts significantly during the sample period

estimate different time-series models before and after the shift

significant seasonality

include seasonal lags

Construction of autoregressive model

Estimate an AR(1) model

Test to see whether the residuals from this model have significant serial correlation. If there is no sequence correlation problem, AR(1) can be used.

If there is a sequence correlation problem, use AR(2) for further estimation and repeat the previous steps. Until there are no sequence problems.

Check for seasonal issues

Method 1: Draw and observe

Method 2: Examine the data to see whether the seasonal autocorrelations of the residuals from an AR model are significant (for example, the fourth autocorrelation for quarterly data)

To correct for seasonality, add seasonal lags to your AR model. For example, if you are using quarterly data, you might add the fourth lag of a time series as an additional variable in an AR(1) or an AR(2) model .

Detecting heteroskedasticity problems conditional heteroskedasticity

ARCH(1)

Regress the squared residual from your time-series model on a labeled value of the squared residual.

Test whether the coefficient on the squared lagged residual differs significantly from 0

If the coefficient on the squared lagged residual does not differ significantly from 0, the residuals do not display ARCH and you can rely on the standard errors from your time-series estimates.

use generalized least squares or other methods to correct for ARCH

out-of-sample forecasting performance

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