Time Series Modeling

Abstract

In this chapter, the reader is introduced to time series modeling. The science (and art) of time series modeling reflects the times series models of Professors Box and Jenkins, Granger, and Hendry. Many economic time series follow near-random walks or random walk with drift processes. This chapter uses the time series modeling of real Gross Domestic product, GDP, as a time series of interest. Time series models can be univariate, where a time series is modeled only by its past values, or multivariate, in which an input series leads an output series, such as a composite index of leading economic indicators, LEI, that can be used as an input to a transfer function model of real Gross Domestic Product, GDP. Economic indicators are descriptive and anticipatory time-series data can be used to analyze and forecast changing business conditions. Cyclical indicators are comprehensive series that are systematically related to the business cycle. Business cycles are recurrent sequences of expansions and contractions in aggregate economic activity. Coincident indicators have cyclical movements that approximately correspond with the overall business cycle expansions and contractions. Leading indicators reach their turning points before the corresponding business cycle turns. The lagging indicators reach their turning points after the corresponding turns in the business cycle.

Appendix: Advanced Time Series Modeling

Time series modeling can involve predicting the conditional mean of a univariate time series and its conditional variance, its second order moment. Volatility forecasting has become extremely important in derivative pricing and risk management. In this chapter the reader is introduced to several techniques used in time series modeling, the univariate generalized autoregressive conditional heteroscedasticity (GARCH) model, developed by Engle [109, 110] and Bollerslev [35], and vector autoregressive models (VAR). In the previous chapter the reader was introduced to the simple autoregressive model, an AR(1):

$$ {x}_t={y}_{t-1}+{\varepsilon}_t $$

(A.1)

If x _t is a stationary time series, such as financial returns, then the series can be expressed as its mean plus a white noise term that is identically and identically distributed, denoted iid v(ε) = σ². If large changes in data are followed by large changes and small changes follow small changes, then there can be volatility clustering, or conditional heteroscedasticity.

A model that allows the conditional variance to be a function of its past history is said to be the bilinear model of Granger and Anderson in Granger and Newbold [140].

$$ {y}_t={\varepsilon}_t{y}_{t-1} $$

(A.2)

The conditional variance is ${\sigma}^2{y}_{t-1}^2$ and can be either zero or infinite. Engle [86] states a preferable model is

$$ {y}_t={\varepsilon}_t{h}_t^{1/2} $$

(A.3)

where ${h}_t={\alpha}_0+{\alpha}_1{y}_{t-1}^2$Equation (A.3) is referred to as the autoregressive conditional heteroscedasticity (ARCH) model.

$$ {y}_t\mid {\psi}_{t-1}\sim N\left(0,{h}_t\right) $$

(A.4)

$$ {h}_t={\alpha}_0+{\alpha}_1{y}_{t-1}^2 $$

(A.5)

$$ {h}_t=h\left({y}_{t-1},{y}_{t-2},\dots, {y}_{t-p},\alpha \right) $$

(A.6)

where p is the autoregressive process and α is the vector of unknown parameters.

$$ {\displaystyle \begin{array}{l}{y}_t\mid {\psi}_{-1}\sim N\left({x}_t\beta, {h}_t\right)\\ {}{h}_t=h\left({\varepsilon}_{t-1},{\varepsilon}_{t-2},\dots, {\varepsilon}_{t-p},\alpha \right)\\ {}{h}_t={h}_s^{.}\left({\varepsilon}_{t-1},{\varepsilon}_{t-2},\dots, {\varepsilon}_{t-p},\alpha \right)\end{array}} $$

(A.7)

$$ {h}_x\left({x}_t,\dots {x}_{t-p}\right) $$

(A.8)

Engle writes an ARCH process as being described in equations (A.4) and (A.6). The conditional variance is

$$ {\sigma}_t^2={E}_{y_t}^2={Eh}_t $$

The general first-order linear ARCH process, with α₀ > 0, is (covariance) stationary if and only if the characteristic equation has all roots outside the unit circle.

$$ {h}_t=\mathit{\exp}\left({\alpha}_0+{\alpha}_1{y_{t-1}}^2\right) $$

(A.9)

$$ {h}_t={\alpha}_0+{\alpha}_1\mid {y}_{t-1}\mid . $$

A pth-order linear case can be written as:

$$ {\displaystyle \begin{array}{l}{y}_t\mid {\psi}_{t-}\sim N\left({x}_1{\beta}_1{h}_t\right)\\ {}{h}_t={\alpha}_0+{\alpha}_1{\varepsilon}_{t-1}^2+\dots +{\alpha}_p{\varepsilon}_{t-p}^2\\ {}{\varepsilon}_t={y}_t-{x}_t\beta \\ {}l=\frac{1}{T}\sum \limits_{t=1}^T{l}_t\end{array}} $$

$$ {l}_t=-{y}_2\log {h}_t-\frac{1}{2}{\varepsilon}_t^2/{h}_t $$

(A.10)

The Generalized Autoregressive Conditional Heteroscedasticity GARCH (p,q) process can be written as:

$$ {\varepsilon}_t\mid {\psi}_{t-1}\sim N\left(0,{h}_t\right) $$

$$ {h}_t={\alpha}_0+{\sum}_{i=1}^q{\alpha}_i{\varepsilon}_{t-1}^2+{\sum}_{i=1}^p{\beta}_i{h}_{t-i} $$

(A.11)

$$ {h}_t={\alpha}_0+A(L){\varepsilon}_t^2+B(L){h}_t $$

where

$$ \mathrm{p}\ge 0,\mathrm{q}>0 $$

$$ {\upalpha}_0>0,{\upalpha}_1\ge 0 $$

$$ {\mathrm{B}}_{\mathrm{i}}\ge 0\kern0.5em \mathrm{i}=1,\dots, \mathrm{p} $$

If p is zero, then the GARCH (p,q) model reduces to ARCH (q) process.

If all roots of 1-β(z) = 0 lie outside the unit series, then

$$ {\displaystyle \begin{array}{c}{h}_t={\alpha}_0{\left(1-B(l)\right)}^{-1}+A(L){\left(1-B(L)\right)}^{-1}{\varepsilon}_t^2\\ {}\begin{array}{l}={\alpha}_0{\left(1-\sum \limits_{i=1}^p{\beta}_i\right)}^{-1}+\sum \limits_{i=1}^{\infty }{\delta}_i{\varepsilon}_{t-i}^2\\ {}{\delta}_i=\sum \limits_{j=1}^n{\beta}_j{\delta}_{t-j}\end{array}\end{array}} $$

(A.12)

The GARCH (p,q) process can be written:

$$ {\displaystyle \begin{array}{l}{\varepsilon}_t^2={\alpha}_0+\sum \limits_{i=1}^q{\alpha}_j{\in}_{t-j}^2+\sum \limits_{j=1}^p{\beta}_j{\varepsilon}_{t-j}^2-\\ {}\sum \limits_{j=1}^p{\beta}_j{v}_{t-j}+{v}_t\end{array}} $$

$$ {v}_t={\varepsilon}_t^2-{h}_t=\left({n}_t^2-1\right){h}_t $$

(A.13)

$$ {n}_t\sim N\left(0,1\right). $$

The GARCH (i,l) process is:

$$ {h}_t={\alpha}_0{\alpha}_1{\in}_t{\varepsilon}_{t-1}^2+\beta {h}_{t-1}, $$

(A.14)

$$ {\displaystyle \begin{array}{c}{\alpha}_0\ge 0,\\ {}{\alpha}_1\ge 0,\\ {}{\beta}_1\ge 0,\end{array}} $$

or

$$ {\sigma}_t^2={\alpha}_0+{\alpha}_1{\varepsilon}_{t-1}^2+{b}_1{\sigma}_{t-1}^2 $$

(A.15)

$$ {\varepsilon}_t^2={a}_0+\left({a}_1+{b}_1\right){\varepsilon}_{t-1}^2+{u}_t-{b}_1{u}_{t-1} $$

(A.16)

$$ {u}_t={\varepsilon}_t^2-{E}_{t-1}\left({\varepsilon}_t^2\right) $$

$$ E\left({\varepsilon}_t^2\right)={a}_0+\left({a}_1+{b}_1\right)E\left({\varepsilon}_{t-1}^2\right) $$

and

$$ E\left({\varepsilon}_t^2\right)={a}_0+\left({a}_1+{b}_1\right)E\left({\varepsilon}_t^2\right) $$

$$ Var\left({\varepsilon}_t\right)=\frac{a_0}{1-\left({\sum}_{i=1}^p{a}_i+{\sum}_{j=1}^q{b}_i\right)} $$

$$ \left({\varepsilon}_t^2-\frac{a_0}{1-{a}_1-{b}_1}\right)=\left({a}_1+{b}_1\right){\varepsilon}_{t-1}^2-\frac{a_0}{1-{a}_1-{b}_1}+{u}_t-{b}_1{u}_{t-1}. $$

(A.17)

$$ \left({\varepsilon}_{t+k}^2-\frac{a_0}{1-{a}_1-{b}_1}\right)={\left({a}_1+{b}_1\right)}^k\left({\varepsilon}_t^2-\frac{a_0}{1-{a}_1-{b}_1}\right)+{n}_{t+k} $$

(A.18)