Forecasting with Time Series Data

Forecasting With Linear Univariate Time Series Data

• Forecasting univariate time series data involves a 7 step process, designed to make accurate predictions based on historical data
• These include testing for seasonality, time trends, stationarity, determining the appropriate model, estimating parameters, checking model adequacy, and forecasting

Step 1: Test for Seasonality

• Many financial time series exhibit seasonal or periodic behavior
• Seasonality describes a time series with a significantly higher autocorrelation between certain time periods
• Monthly data is often seasonal with a periodicity of 12, demonstrated by peaks in the same month each year
• Quarterly data is seasonal with a periodicity of 4, as demonstrated by peaks in the same quarter of each year
• Seasonal patterns need to be removed before other time series analysis
• Removing seasonality is referred to as seasonal adjustment

Detecting Seasonlity

• Obvious periodic humps may be seen by plotting the original data
• ACFs may indicate seasonality, as they may include higher autocorrelation at fixed lag intervals
• Other statistical methods like Chi-square and Kolmogorov-Smirnov tests may be used

Detecting Time Trend Adjustments

• Seasonal differencing can be used to eliminate seasonality, by removing the high autocorrelation
• Seasonally differencing a time series with periodicity s, and seasonal adjustment is required
• For monthly time series, seasonal adjustment means using the year-over-year change
• The year-over-year change calculates the true change by taking the difference between the same month of different years
• Normal month-to-month changes do not consider seasonality
• Adjustments can also be made to quarterly time series

Step 2: Test for Time Trend

• Graph the data over time to determine if there is a time trend

Linear Time Trends

• Indicate a linear relationship between time and the data

NonLinear Time Trends

• Includes different cases for variables in logarithms
• Case 1: Linear-log model with X in logarithms, Y is not
• Case 2: Log-linear model with Y in logarithms, X is not
• Case 3: Log-log model with both X and Y in logarithms

Step 3: Test For Stationarity

• Process requires testing for stationarity to ensure the data's statistical properties don't change over time

Weakly Stationary Data

• The 2nd order weakly stationary features:
  • Constant Mean
  • Constant Variance
  • Time invariant Covariance, depends on the difference between 2 periods

Nonstationary Situations

• Stationarity requires that the absolute values of the solutions of the characteristic equation are greater than 1

The First Situation

• This situation is uncommon
• The nonstationary situation rarely happens

The Second Situation

• This situationis more likely to happen
• Under the most likely nonstationary situation, all the ACFs are equal to 1
• Called a Unit Root
• Unit root is equivalent to non-stationarity

Methods of Detecting Non-Stationarity

• Visually inspect the data for trends in changing mean or variance
• Inspect the ACFs of the data
• Augmented Dickey-Fuller tests can detect unit roots, indicating non-stationarity

Augmented Dickey-Fuller Unit Root Test

• If the t-statistic is less than critical values, the null hypothesis can be rejected
• Conclude that the data is stationary

Dickey-Fuller Test For AR(1)

• This test confirms non stationarity
• Testing, as the data is considered nonstationary is equivalent to testing
• Can measure if data is considered in alternative or null hypothesis

Augmented Dickey-Fuller Unit Root Test Details

• Augmented tests add additional difference terms
• Can be applied for an AR(1), AR(2), AR(k)

Hypotheses

• The null hypothesis, suggesting the data has a unit root and is nonstationary
• The alternative hypothesis, suggesting the data is stationary
• The ADF test is a one-tail test
• The t-statistic compares with critical values, used to make the rejection decision

Dealing With Stationarity

• If the unit root test shows the data is stationary, fitting it into a univariate time series model can be the right call
• If data isn't stationary, the first difference needs to be used
• First difference is used until the data shows high autocorrelation

Differenced Data

• You test the stationarity of the first differenced data
• The first-differenced data can be fit into a time series model, should it be stationary
• The difference may need to be taken many times if the data continues to be not stationary

Step 4: Determine The Appropriate Univariate Time Series Model

• An AR, MA, or ARMA is determined by using the ACF and PACF

Standards

• AR(k) models have gradually decaying ACFs and PACFs that cut off at lag k
• MA(q) models have PACFs that decay gradually and ACFs that cut off at lag q
• ARMA (k, q) models will have ACFs and PACFs which both decay gradually, with no cutoff

More Details For Model Selection

• Use an AR(k) model if the ACF decays and the PACF cuts off at lag k
• PACF helps figure out the i.e. value of k, in the AR model
• This model would be something like AR(3)
• The MA(q) model should be used if the ACF cuts off at lag q and the PACF decays gradually
• ACF helps determine the value of q) in the MA model if the ACF will cut off at lag 3
• Use an ARMA model comes into play if neither ACF or PACF cuts off, but both decay gradually
• The number of Lags can be defined by the AIC or BIC algorithms

ARIMA(k,i,q)

• ARMA has unit root is called an ARIMA(k,i,q) model
• Unit-root nonstationary ARIMA becomes transformed into a stationary model by using first differences
• ARMA is equivalent to ARIMA
• AR(1) is close to ARIMA(1,0,0)

Step 5: Determining Parameters

• Include conditional-sum-of-squares, max likelihood estimation and combinations of them

Step 6: Evaluate Adequacy

• Confirms the estimated model is correct
• If the model is adequate the residual model is a white noise

Inadequacy

• If the model is inadequate, the residual is not white noise
• Other models need to be used, consider adding more lags

Evaluate The Estimated Model

• Method 1: Check the ACF of the residual
• This is related to detecting if the estimated model is adequate
• Method 2: Implement the Ljung-Box test
• Called the portmanteau text
• Method 3: Implement the Durbin-Waston test
• The Ljung-Box model tests autocorrelation with a chi squared distribution
• Includes the term in the first m lags, which follows a chi-squared distribution
• There is no autocorrelation
• There exists autocorrelation if the model is inadaquate

Performing The Ljung-Box Test

• Reject the null for the alternative value, if the p-value is over 0.05 it can be considered inadequate
• If the p-value is greater than 0.05, do not reject the null hypothesis, then it can be considered adequate
• Special as it is not desired to reject the null hypothesis throughout the Ljung-Box test

Explores forecasting univariate time series data using a 7-step process. Covers testing for seasonality, time trends, and stationarity. Also looks at model determination and seasonal adjustments.