Time Series

Question 1

Time Series forecasting steps

Answer 1

Question 3

Time Series Models:

1. Linear
2. Log-linear
3. Autoregressive (AR)

Answer 3

Linear Trend

y_t = b₀ + b₁ (t) + e_t

• predicted change in y is b₁ and t = 1, 2, …, T
• uses time as independent variable (model assumes it explains dependent variable which limits it)
• DW check for SC.
• Appropriate for data points equallly distributed above/below line w/ constant mean. GDP and inflation good candidates for model.

Log-linear

y_t = e^{b₀ + b₁ (t)}

ln( y_t ) = ln(e^{b₀ + b₁ (t)}) => ln( y_t ) = b₀ + b₁ (t)

• best when data residuals are correlated/predictable, or mean is non-constant. Investment and seasonality data good candidates. Expotential growth data needs it.
• increases predictive ability of time series and minimizes impact of SC in error terms.

Autoregressive (AR)

x_t = b₀ + b₁ x_t-1 + b₂x_t-2 + …. + b_px_t-p + e_t

• above shows AR model of the order p
• AR(p) model is correctly specified if autocorrelations of residuals from model are not statistically significant at any lag.
• No longer a distinction b/w dependent and independent variables ( x is the only variable)
• Don’t use Durbin-Watson here.
• Do use t-test to find if residuals at any lag are significant. If yes, model incorrect and lagged variable at indicated lag should be added.
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Question 4

Time Series terms (1st set)

Chain rule of forecasting

Covariance stationary

Mean Reversion

Unit Root

Random Walk

Answer 4

Chain rule of forecasting - calculation of successive forecasts one period at a time, where risk increases w/ each forecast bc based on previously forecasted values

^x_t+1 = ^b₀ + ^b₁x_t is one-period ahead for an AR(1). x_t+2 is two-periods.

Covariance stationary - infrences based on AR models may be invalid unless model is covariance stationary, meaning the following three conditions are met:

Constant and finite

1. Mean
2. Variance
3. Covariance w/ leading or lagged values

Most models are not stationary.

How to determine?

• plot data to see if mean and variance remain constant
• (ryan) Dickey-Fuller test (test for unit root, or if b₁ - 1 = 0)

Mean Reversion mean reverting level for an AR(1) is:

b₀ / (1 - b₁ )

Unit Root present when lag coefficient = 1.

• Not covariance stationary
• Undefined mean reversion (i.e. b₁ = 1)
• period’s value = last period’s value + random error term

Random Walk

Without Drift: y_t = x_t-1 + E_t

• (Same descripters as unit root) when the value in one period is equal to the value in another period, plus a random (unpredictable) error.
• First Differencing can transform data to covariance stationary

First Differencing is subtracting value of immediately preceding period from current period value to define a new variable, y. Models change in the value of the variable rather than the value of the variable.

y_t = x_t – x_{t – 1} ⇒ y_t = ε_t

Stating y in form of AR(1) model:

y_t = b₀ + b₁y_{t – 1} + ε_t

where:

b₀ = b₁ = 0

Question 5

Time Series terms (2nd set)

Seasonality

Root Mean Squared Error (RMSE)

Autoregressive Conditional Heteroskedasticity (ARCH)

Answer 5

Seasonality

Root Mean Squared Error (RMSE)

Autoregressive Conditional Heteroskedasticity (ARCH)

• What is it? variance of the residuals in one time period within a time series is dependent on the variance of the residuals in another period
• Effect? standard errors of regression coefficients in AR models and the hypothesis tests of these coefficients are invalid.
• Correct? use methods that correct for heteroskedasticity, such as generalized least sqaures.
  
  • ​Alternatively, using an ARCH modelcan be used to predict variance in t+1