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Introduction to Time Series Analysis

Regression vs. Time Series Analysis

- In regression analysis, we estimate models that

attempt to explain the movement in one variable

by relating it to a set of explanatory variables - Time series analysis attempts to identify the

properties of a time series variable and use

models to predict the future path of the variable

based on its past behavior - Example How do stock prices move through time

Fama (1965) claimed that they identify with the

random walk process

Regression vs. Time Series Analysis

- Multiple regression analysis with time series

data can also lead to the problem of spurious

regression - Example Suppose we estimate the following model

with time series data - The estimated regression may turn out to have a

high R-sq even though there is no underlying

causal relationship - The two variables may simply have the same

underlying trend (move together through time)

A Simple Time Series ModelThe Random Walk Model

- How can we model the behavior of financial data

such as stock prices, exchange rates, commodity

prices - A simple model to begin with is the random walk

model given by - This model says that the current value of

variable y depends on - The variables value in the previous period
- A stochastic error term, which is assumed to have

mean zero and a constant variance

A Simple Time Series ModelThe Random Walk Model

- What does this model imply about a forecast of a

future value of variable y - According to the model
- Therefore, the expected future value of variable

y is - given that the expected value of the error term

is zero

A Simple Time Series ModelThe Random Walk Model

- Implication The best forecast of the future

value of variable y is its current value - If variable y follows a random walk, then it

could move in any direction with no tendency to

return to its present value - If we rewrite the random walk model as follows
- then we refer to a random walk with a drift,

meaning a trend (upward or downward)

White Noise Process

- Suppose that the variable y is modeled as follows
- where t is a random variable with mean zero,

constant variance and zero correlation between

successive observations - This variable follows what is called a white

noise process, which implies that we cannot

forecast future values of this variable

Stationarity in Time Series

- In time series analysis, we attempt to predict

the future path of a variable based on

information on its past behavior, meaning that

the variable exhibits some regularities - A valuable way to identify such regularities is

through the concept of stationarity - We say that a time series variable Yt is

stationary if - The variable has a constant mean at all points in

time - The variable has a constant variance at all

points in time - The correlation between Yt and Yt-k depends on

the length of the lag (k) but not on any other

variable

Stationarity in Time Series

- What type of a time series variable exhibit this

behavior - A variable that moves occasionally away from its

mean (due to a random shock), but eventually

returns to its mean (exhibits mean reversion) - A shock in the variable in the current period

will be reflected in the value of the variable in

future periods, but the impact diminishes as we

move away from the current period - Example The variable of stock returns of Boeing

exhibits the properties of stationarity

Boeings monthly stock returns (1984-2003)

Stationarity in Time Series

- A variable that does not meet one or more of the

properties of stationarity is a nonstationary

variable - What is the implication of nonstationarity for

the behavior of the time series variable - A shock in the variable in the current period

never dies away and causes a permanent deviation

in the variables time path - Calculating the mean and variance of such a

variable, we see that the mean is undefined and

the variance is infinite - Example The SP 500 index (as opposed to the

returns on the SP index which exhibit

stationarity)

The SP 500 Index Exhibits Nonstationarity

The Returns on the SP 500 Exhibit Stationarity

The Impact of Nonstationarity on Regression

Analysis

- The major impact of nonstationarity for

regression analysis is spurious regression - If the dependent and explanatory variables are

nonstationary, we will obtain high R-sq and

t-statistics, implying that our model is doing a

good job explaining the data - The true reason of the good model fit is that the

variables have a common trend - A simple correction of nonstationarity is to take

the first differences of variables (Yt Yt-1),

which creates a stationary variable

Testing for Nonstationarity

- A common way to detect nonstationarity is to

perform a Dickey-Fuller test (unit root test) - The test estimates the following model
- and test the following one-sided hypothesis

Testing for Nonstationarity

- If the estimate of 1 is significantly less than

zero, then we reject the null hypothesis that

there is nonstationarity (meaning that variable Y

is stationary) - Note The critical values of the t-statistics for

the Dickey-Fuller test are considerably higher

than those in the tables of the t distribution - Example For n 120, the critical t-statistic

from the tables is near 2.3, while the

corresponding value from the Dickey-Fuller tables

is 3.43

Characterizing Time Series VariablesThe

Autocorrelation Function (ACF)

- The ACF is a very useful tool because it provides

a description of the underlying process of a time

series variable - The ACF tells us how much correlation there is

between neighboring points of a time series

variable Yt - The ACF of lag k is the correlation coefficient

between Yt and - Yt-k over all such pairs in the data set

Characterizing Time Series VariablesThe

Autocorrelation Function (ACF)

- In practice, we use the sample ACF (based on our

sample of observations from the time series

variable) to estimate the ACF of the process that

describes the variable - The sample autocorrelations of a time series

variable can be presented in a graph called the

correlogram - The examination of the correlogram provides very

useful information that allows us to understand

the structure of a time series

Characterizing Time Series VariablesThe

Autocorrelation Function (ACF)

- Example Does the ACF of a stationary series

exhibit a certain pattern that can be detected by

studying the correlogram - For a stationary series, the autocorrelations

between two points in time, t and tk, become

smaller as k increases - In other words, the ACF falls off rather quickly

as k increases - For a nonstationary series, this is usually not

the case, as the ACF remains large as k increases

Correlogram and ACF of SP Index Variable

- Note that as the number of lags (k) increases,

the ACF declines, but at a very slow rate - This is an indicator of a nonstationary variable
- Compare this result with the graph of the level

of the SP Index shown previously

Correlogram and ACF of Returns on the SP Index

- An examination of the correlogram of the variable

of returns on the SP index shows that this

variable exhibits stationarity - The ACF declines very rapidly, meaning that there

is very low correlation between observations in

periods t and tk as k increases

Characterizing Time Series VariablesThe

Autocorrelation Function (ACF)

- To evaluate the quality of information from the

correlogram, we assess the magnitudes of the

sample autocorrelations by comparing them with

some boundaries - We can show that the sample autocorrelations are

normally distributed with a standard deviation of

1/(n)1/2 - In this case, we would expect that only 5 of

sample autocorrelations would lie outside a

confidence interval of 2 standard deviations

Characterizing Time Series VariablesThe

Autocorrelation Function (ACF)

- Given that the correlogram shows values of

autocorrelations, these values cannot lie outside

the interval 1 - As the number of time series observations

increases above 40-50, the limits of the

confidence interval given by the standard

deviations become smaller - In practical terms, if the sample

autocorrelations lie outside the confidence

intervals given by the correlogram, then the

sample autocorrelations are different from zero

at the corresponding significance level

Correlograms and Confidence Intervals for Sample

Autocorrelations

From Sample Data to Inference About a Time Series

Generating Model

Sample Data

Sample Autocorrelations

Population Autocorrelation

Generating Model

Linear Time Series Models

- In time series analysis, the goal is to develop a

model that provides a reasonably close

approximation of the underlying process that

generates the time series data - This model can then be used to predict future

values of the time series variable - An influential framework for this analysis is the

use the class of models known as Autoregressive

Integrated Moving Average (ARIMA) models

developed by Box and Jenkins (1970)

Autoregressive (AR) Models

- In an AR model, the dependent variable is a

function of its past values - A simple AR model is
- This is an example of an autoregressive model of

order 1 or an AR(1) model - In general, an autoregressive model of order p or

AR(p) model will include p lags of the dependent

variable as explanatory variables

Autoregressive (AR) Models

- Is it possible to conclude that a time series

follows an AR(p) model by looking at the

correlogram - Example Suppose that a series follows the AR(1)

model - The ACF of the AR(1) model begins with the value

of 1 and then declines exponentially - The implication of this fact is that the current

value of the time series variable depends on all

past values, although the magnitude of this

dependence declines with time

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