Quant Methods #11 - Time-Series Analysis

Question 1

types of time series models

Answer 1

LOS 11.a

Trend Models

• linear: y_t = b₀ + b₁t + e_t
• log-linear: y_t = e^b₀+b₁t ⇒ ln(y_t) = b₀ + b₁t + e_t

Lagged Models

• autoregressive AR(p): x_t = b₀ + b₁x_t-1 + b₂x_t-2 + … + b_px_t-p + e_t

Question 2

serial correlation in Trend model

Answer 2

LOS 11.b

One assumption of linear regression is that regression residuals are uncorrelated with each other; violation of this assumption is called autocorrelation or serial correlation.

• use Durbin Watson to test for serial correlation
• try log-linear transform of time series data to remove it
• if log-linear model still exhibits autocorrelation, try using autoregression model

Question 3

covariance stationary

Answer 3

LOS 11.c

To use AR models, time series must be covariance stationary. A time series is covariance stationary if all 3 of these are true:

• constant & finite expected values (i.e. mean-reverting level)
• constant & finite variance
• constant & finite covariance between values at any given lead or lag

if not covariance stationary time series, AR model will produce meaningless results

Question 4

serial correlation in AR model

Answer 4

LOS 11.e

x_t = b₀ + b₁x_t-1 + e_t

• cannot use Durbin Watson to test for serial correlation in AR models
• use a t-test on residual autocorrelations; if serial correlation exists, then model is incomplete
• solution: add more lagged variables e.g. AR(1) → AR(2)

Question 5

testing for serial correlation in AR model (used to test the “fit” of the autoregressive model)

Answer 5

LOS 11.e

1. estimate AR model using linear regression; start with AR(1)
2. calculate autocorrelation of residuals (i.e. e_t-e_t-1, e_t-e_t-2, etc)
3. test for autocorrelation significance:

• t-test; H₀: p_{e(t), <em>e</em>(t-k)} = 0
• t-stat = p_{e(t), <em>e</em>(t-k)} / [1/sqrt(T)], df = T - 2

Question 6

seasonality

Answer 6

LOS 11.l

seasonality - a time series showing consistent seasonal patterns

detecting seasonality - autocorreltaion of residuals will show strong correlation (i.e. high t-stats) to certain lags

solution - bring seasonal lag into the model

Question 7

mean reversion of time series

Answer 7

LOS 11.f

a time series is mean-reverting if value of dependent variable tends to move towards its mean (i.e. ^x_t = x_t-1)

the mean-reverting level for AR(1) is

MRL = x_t = b₀ / (1 - b₁)

NOTE: all covariance stationary time series have a finite mean-reverting level. For AR(1) this means that |b₁| < 1

Question 8

comparing model accuracy of models

Answer 8

LOS 11.g

in-sample forecasts

• forecasts (^y_t) are within the time series data sample range
• forecast accuracy: compare errors (y_t - ^y_t)

out-of-sample forecasts

• forecasts (^y_t) are outside the time series data sample range
• forecast accuracy: compare root mean squared errors (RMSE)

The model having lowest RMSE for out-of-sample data will have better predictive power in the future

Question 9

regression coefficient instability

Answer 9

LOS 11.h

• instability (aka nonstationarity) - estimated regression coefficients can change (degrade) over time
• trade-off between statistical reliability of long time series and stability of short time series
• has economic progress or environment changed? If “yes” then historical data may not produce a reliable model

Question 10

unit roots (def)

Answer 10

LOS 11.j,l

• In an AR(1) model, coefficient must be < 1.0
• if coefficient = 1 ⇒ unit root (i.e. b₁ = 1)
• unit root is defining feature of random walks
• common in series that consistently increase or decrease over time (e.g. 1990’s stock market)
• Two types of random walks: “drift” and “no drift”
  
  • w/o drift: x_t = 0 + 1x_t-1 + e_t (note b₀ = 0)
  • with drift: x_t = b₀ + 1x_t-1 + e_t, b₀ != 0

Question 11

unit roots (impact, detection, correction)

Answer 11

LOS 11.i-k

impact: unit root models are not covariance stationary (i.e. “nonstationarity”)

MRL = b₀ / (1 - b₁) = b₀ / 0 = infinity

detection: use Dickey-Fuller test for a unit root

correction: use first differencing to remove unit roots

Question 12

Dickey-Fuller Test for a unit root

Answer 12

LOS 11.j-k

• transform AR(1) model: subtract xt-1 from both sides

x_t - x_t-1 = b₀ + (b₁ - 1)x_t-1 + e_t
= b₀ + g₁x_t-1 + e_t, where g₁ = (b₁ - 1)

if g₁ = 0, then there is a unit root in AR(1) model

• DF test:
  
  • H₀: g₁ = 0, times series has a unit root and is therefore nonstationary
  • H_a: g₁ < 0, no unit root in time series
  • modified t-test for g₁ with special t_c table
  • if null is rejected, , then no unit root

Question 13

first differencing

Answer 13

LOS 11.i-k

• use First Differencing to remove unit roots
• create a new dependent variable, y, defined as the change in x:
  
  • y_t = x_t - x_t-1 = e_t ⇒ y_t = b_{<span>0</span>} + b₁y_t-1 + e₁, b₀ = b₁ = 0

Question 14

autoregressive conditional heteroskedasticity (ARCH)

Answer 14

LOS 11.m

For a single times series such as an AR model, ARCH exists when the variance of residuals in one period is dependent (i.e. a function of) on the variance of the residuals in a previous period (for ARCH(1) model, the most previous period).

impact: causes incorrect standard errors of the coefficients in AR models, which makes the hypothesis tests of these coefficients invalid

detection: use ARCH(1) regression of squared residuals

solution: use generalized least squares (or other method) to correct for heteroskedasticity

Question 15

testing for ARCH

Answer 15

LOS 11.m

To test for ARCH, use ARCH(1) model that regresses the squared residuals on the first lag of the squared residuals:

ê_t² = a₀ + a₁ê_t-1² + u_t,

where a₀ is a constant and u_t is an error term

if the coefficient a₁ is statistically different from zero, then the time series is ARCH(1)

Question 16

time series construction flowchart

Answer 16

LOS 11.o

Question 17

multiple time series

Answer 17

LOS 11.n

Before regressing two time series (e.g. y_t = b₀ + b₁x_t + e_t), check for covariance nonstationary (unit roots) on both time series (separate DF tests for each).

• both are covariance stationry → OK
• only one is covariance stationry → Not reliable
• neither covariance stationry → OK if cointegrated

Question 18

cointegration

Answer 18

LOS 11.n

cointegration - two time series are related to the same macro variables or follow the same trend

if two time series are cointegrated, the residual terms from regressing the two on each other is covariance stationary and t-tests are therefore reliable.