ChVIIntroductiontoTimeSeries.pptx

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VI. Introduction to Time Series – AR(1) Model and Forecasting

a. Introduction to Dependent Observations b. Checking for Independence

c. Autocorrelation d. The AR(1) Model e. Random Walks f. Trend Models

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a. Introduction to Dependent Observations

In autoregressive models, we consider observations taken over time.

To denote this, we will index the observations with the letter t rather than the letter i.

Our data will be observations on Y₁, Y₂, ...Y_t, ...where t indexes the day, month, year, or any time interval.

Key new idea:

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We will NOT assume that Y_t-1 is independent of Y_t

Example: Is tomorrow’s temperature independent of today’s?

Suppose y₁ ...y_T are the temperatures measured daily for several years. Which of the following two predictors would work better:

i. the average of the temperatures from the previous year ii. the temperature on the previous day?

If the readings are iid N(μ, σ2), what would be your prediction for

Y_T+1?

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Monthly US Beer Production (millions of barrels)

Strong

Seasonality

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b. Checking for Independence

It is not always easy just to look at the data and decide whether a time series is independent.

So how can we tell?

Plot Y_t vs. Y_t-1 to check for a relationship or

Plot Y_t vs. Y_t-s for s = 1, 2, …

Knowing Y_t does not help you in predicting Y_t+1

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How do we do this in R? – Use the “back” command

18.22 18.19 6 17.62 18.22 5 17.65 17.62 4 15.00 17.65 3 15.19 15.00 2 * 15.19 1 b_prod(t-1) b_prod

Now each row has Y at time t, and Y one

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Now let’s return to the lake data…

First, let’s plot Level_t vs. Level_t-1 Corr = .794

Each point is a pair of adjacent years. e.g. (Level₁₉₂₉, Level₁₉₃₀)

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c. Autocorrelation

Time series is about dependence. We use correlation as a measure of dependence.

Although we have only one variable, we can compute the correlation between Y_tand Y_t-1 or between Y_tandY_t-2.

The correlations between Y’s at different times are called

autocorrelations.

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We will assume what is known as stationarity.

Roughly speaking this means:

– _{The time series varies about a fixed mean and has constant}

variance

– _{The dependence between successive observations does}

not change over time

Let’s define the autocorrelations for a stationary time series.





 











 





 





t t s t t s

s

t

t t s

cov Y ,Y

Var Y

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We estimate the theoretical quantities by using sample averages (as always).

The estimated or sample autocorrelations are:

  





T

t t s t s

s _T

2 t

t 1

(Y

Y)(Y

Y)

=

r

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The ACF command in R computes the autocorrelations

There is a strong dependence

between

observations spaced close together in

time (e.g only one or two years apart). As time passes, the

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Let’s look at the autocorrelations for the IID series.

In contrast to the ACF for the

‘level’ series, the sample

autocorrelations are much

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How do we know if the sample autocorrelations are good estimates of the underlying theoretical autocorrelations? and

How do we know if we have enough sample information to reach definitive conclusions?

If all the true autocorrelations are 0, then the standard

deviation of the sample autocorrelations is about 1/sqrt(T).

 

T

1

r

Err

Std

_s



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For the IID series

All of the sample autocorrelations are within 2 standard

deviations of 0 -- no evidence of positive autocorrelation in the data.

For the level series

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d. The AR(1) Model

A simple way to model dependence over time is with the “autoregressive model of order 1.”

This is a SLR model of Y_t regressed on lagged Y_t-1.



   

 

t 0 1 t 1 t

AR(1) : Y

Y

What does the model say for the T+1 st observation?

1 T T 1 0 1 T

Y

_



β



β



ε

_

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How should we predict Y_T+1?



T 1 T



  

0 1 T

 



T 1 T



  

0 1 T

E Y |Y

Y

E

|Y

Y

How do we use the AR(1) model? We simply regress Y on lagged Y.

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Now let’s look at the ACF of the residuals…

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Now let’s look at the ACF of the residuals…

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To gain a better feel for this model, let’s simulate data series from the model with various parameter settings…

The series fluctuates around a mean level with fairly long “runs”. 0

1

0

.8

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Now let’s look at a series generated with a negative slope value…

Because β₁ is negative, an above average Y tends to be followed by a below average Y (and vice versa) - hence the jagged

appearance of the plot.

0

1

0

.8

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Now let’s look at a series generated with a slope value of 1…

Wanders around quite a lot! Can exhibit what appears to be trends.

0

1

0

1.0

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Some Intuition on Mean Reversion

We have seen that the slope parameter governs the rate at which the AR(1) model “returns” or “reverts” to the mean level of the series.

Fact for the AR(1) model:

 

  







 

0 t

1

E Y

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If we subtract μ from both sides of the AR(1) model equation, we can write the model in terms of deviations from the mean.



_t ₁



_t

1

t

Y



μ



β

_



μ



ε

Thus, β₁ governs the rate at which you “revert” to the mean level of the series.

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e. Random Walks

The case of β₁ = 1 deserves special attention because of it's importance in economic data series.

Many economic and business time series display a "random walk character."

A random walk is an AR(1) model with β₁ = 1

Random Walk:

t 1

t o

t

Y

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e. Random Walks

The intercept, β₀ , is called the drift parameter for the random walk. Let's first consider the case of β₀ = 0.

This called a random walk with zero drift:

t 1

t

Y



_



ε

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e. Random Walks

The random “walk” get its name from the idea of a random walker on the number line. A random walker is someone who has an equal chance of taking a step forward or a step backward. The size of the steps are random as well.

To see this, it is very useful to re-express the random walk in term of increments or steps. Subtract Y_t-1 from both sides,

The increments are an random sample (iid collection of rvs)! t

1 t t

t

Y

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e. Random Walks

A random walk with zero drift:

– _{"meanders" around zero with no particular trend.}

– _{can take long "excursions" away from zero that look like}

trends. Don’t get fooled!

– _{but random walk will always return to zero.}

Random Walks with Non-zero Drift:

If β₀ is positive, we have a random walk with positive drift.

Here the average step size is β₀ :

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f. Trend Models

Many times we want to allow for shifts in the mean of series over time.

Linear Trend Model:

50 40 30 20 10 30 20 10 0 Index C 3 50 40 30 20 10 30 20 10 0 Index C 3 t 1 0 t

t

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f. Trends and Random Walks

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f. Random Walks and Trends

Let’s run the regression for the trend fit and look at residual acf. Looks pretty auto-correlated! Trend Model is not

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f. Random Walks and Trends

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g. Stock Prices and Market Efficiency

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It turns out that a model that fits many stock prices series is a random walk in the log of prices.

log p

  

_t

α

 log p



_t₋₁

 

ε

_t

o

log p

 

_t

−

log p



_t₋₁

 

α



ε

_t

log 1



Δ

p

t

p

_t₋₁

⎛

⎝⎜

⎞

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Thus, if the log of stock prices follows a random walk, the changes in the log of the price are independent.

This has profound implications for the ability to predict future changes in stock prices. This says that stock price changes are completely independent of past changes.

This strongly suggests that any trading strategy that involves the past history of price changes can’t work (momentum

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What is the economic meaning of this finding?

One possible explanation is that stock prices reflect all available information at the time of trade. Competitive markets provide a sort of information aggregation

mechanism by which information relevant to the stock price (e.g. future profitability of the firm) is incorporated into price.

This idea is often called the weak market efficiency hypothesis. This idea goes back to the fundamental principle of conditional prediction – i.e. forecast errors

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h. Example of a Time Series Regression

With time series data, we want to use information available now (time t) to forecast some variable of interest in the

future. In the AR(1) model, we use the current value of y to forecast future values of y.

We might wish to use some other time series to forecast the future value of variable of interest. A nice example of this is the relationship between sales and advertising.

data(persistence)includes data on monthly sales ($)



t 1 t



   

0 1 t

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Peaks at the quarterly frequency.

Let’s model this with lags at frequencies related to

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We have three lags, a semi annual and an annual effect.

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Residuals are pretty clean – other than at lag 1, we see little

auto-correlation.

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Wrong way: model using only lags of ADVINF!

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h. Carry-over Effects of Advertising

In these models, advertising has an immediate effect and then a carry-over effect from the lagged sales terms. These terms can proxy for repeat sales, word-of-mouth, etc.

Consider a simple model:

There is an immediate effect of δ and a carry-over effect of λ.

The carry-over effect means that there is a larger long-run effect of advertising that the short-run effect. Let’s trace the

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t  0 δ

t 1 λ × δ

t 2 λ × λ × δ

M M

t=s λs ×δ

long-ru n: δ₍1+λ +λ2+λ3+K₎= 1 1−λ ( )δ

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Let’s write the model down.

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Glossary of Symbols

ρ_s - sth order autocorrelation

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Important Equations





 











 





 





t t s t t s

s

t

t t s

cov Y ,Y

Var Y

  





T

t t s t s

s _T

2 t

t 1

(Y

Y)(Y

Y)

=

r

(Y

Y)

 

T

1

r

Err

Std

_s



Population and Sample Autocorrelations

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Important Equations



   

 

t 0 1 t 1 t

AR(1) : Y

Y



_t ₁



_t

1

t

Y



μ



β

_



μ



ε

t 1

t o

t

Y



β



_



ε

definition of AR(1) model

Mean Reversion form of AR(1)

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Glossary of R Commands

• acf()_{: Computes (and by default plots) estimates of the}

autocorrelation function

• back()_{: Computes a lagged version of a time series,}

shifting the time base back once.

• diff()_{: Returns the differences between a value and its}

lagged value.

• c(1:30)_{: Generates 30 numbers from 1 to 30 with}