31 terms

time series

trend component, seasonal component, cyclical component, irregular component

cointegrated

Two series are said to be ___________________ if the series do not diverge from each other without bound in the long-term.

time series data

Data on the outcome of a variable or variables in different time periods

ex: the quarterly sales for a company for the past 5 years

Time-series data are prevalent in finance and can be challenging because they are likely to violate the underlying assumptions of linear regression

ex: the quarterly sales for a company for the past 5 years

Time-series data are prevalent in finance and can be challenging because they are likely to violate the underlying assumptions of linear regression

two functions of time series

to explain the past and to predict the future in terms of something like sales

linear trend

in which the dependent variable changes at a constant rate over time.

Yt = B0 + B1(t) + Et

Yt = B0 + B1(t) + Et

Log linear trend analysis

in which the dependent variable changes at an exponential rate over time or constant growth at a particular rate

Yt = B0 + B1(t) + Et

e^Yt = the answer

Yt = B0 + B1(t) + Et

e^Yt = the answer

log vs linear

-Is the estimated relationship persistently above or below the trend line?

-Are the error terms correlated?

-We can diagnose these by examining plots of the trend line, the observed data, and the residuals over time.

-Are the error terms correlated?

-We can diagnose these by examining plots of the trend line, the observed data, and the residuals over time.

durbin watson

test whether there is serial correlation

trend models are likely to exhibit serial correlation (autocorrelation)

trend models are likely to exhibit serial correlation (autocorrelation)

autoregressive model

a time series that is regressed on its own past values

notation = Xt

X𝑡= 𝑏0 + 𝑏1𝑥𝑡−1+ 𝑏2𝑥𝑡−2 + ε𝑡

notation = Xt

X𝑡= 𝑏0 + 𝑏1𝑥𝑡−1+ 𝑏2𝑥𝑡−2 + ε𝑡

autoregressive model

AR(p) model, the p indicates how many lagged values of the dependent variable are used

can use longer interval differences to account for seasonality

A correctly-specified autoregressive model has residual autocorrelations that do not differ significantly from zero.

can use longer interval differences to account for seasonality

A correctly-specified autoregressive model has residual autocorrelations that do not differ significantly from zero.

covariance stationary

A time series is said to be _______ if its mean and variance do not change over time.

If a time series is not covariance stationary, linear regression estimates are not valid and have no economic meaning.

If a time series is not covariance stationary, linear regression estimates are not valid and have no economic meaning.

covariance stationary

For a time series to be _______,

1. The expected value of the series must be finite and constant across time.

2. The variance of the series must be finite and constant across time.

3. The covariance of the time series with itself must be finite and constant for all intervals over all periods across time.

1. The expected value of the series must be finite and constant across time.

2. The variance of the series must be finite and constant across time.

3. The covariance of the time series with itself must be finite and constant for all intervals over all periods across time.

mean reverting

if its values tend to fall when they are above the mean and rise when they are below the mean.

In finance, prices and returns, among other variables, tend to move back towards the mean or average.

In finance, prices and returns, among other variables, tend to move back towards the mean or average.

chain rule of forecasting

a process in which the next period's value, predicted by the forecasting equation, is substituted into the equation to give a predicted value two periods ahead.

more you forecast the greater uncertainty or likeliness of error

more you forecast the greater uncertainty or likeliness of error

In sample forecast errors

are the residuals from a fitted time series

ex: using data from Jan 1984 - Dec 2015

_____ are the residuals from Jan 1984 - Dec 2015

ex: using data from Jan 1984 - Dec 2015

_____ are the residuals from Jan 1984 - Dec 2015

out of sample forecast errors

are the difference between predicted values from outside the sample period and the actual values once realized.

root mean squared error

measures model accuracy, We calculate the root mean squared error by first calculating all the errors, square them, calculate the average, and then take the square root of that average.

Coefficient instability

Time-series coefficient estimates can be unstable across time. Accordingly, sample period selection becomes critical to estimating valuable models.

This instability can also affect model estimation because changes in the underlying time-series process can mean that different time-series models work better over different time periods.

EX: an AR(1) model may suit one period well but an AR(2) model may be better for another period

This instability can also affect model estimation because changes in the underlying time-series process can mean that different time-series models work better over different time periods.

EX: an AR(1) model may suit one period well but an AR(2) model may be better for another period

avoiding instability

There are no clear-cut rules for selecting an appropriate time frame for a particular analysis.

Rely on basic sampling theory:

1. Don't use two clearly different populations.

2. Rely on basic time-series properties

3.. Don't mix stationary and nonstationary series or series with different mean or variance terms.

4. The longer the sample period, the more likely the samples come from different populations.

Rely on basic sampling theory:

1. Don't use two clearly different populations.

2. Rely on basic time-series properties

3.. Don't mix stationary and nonstationary series or series with different mean or variance terms.

4. The longer the sample period, the more likely the samples come from different populations.

random walk

is a time series in which the value of the series in one period is the series in the previous period plus an unpredictable random error term

AR(1) series where B0=0 and B1=1

the best predition of tomorrow is the value today plus a random error term

AR(1) series where B0=0 and B1=1

the best predition of tomorrow is the value today plus a random error term

random walk

The theory is that stock price changes follow ______. That is, stock price changes are random and unpredictable.

mean reversion is undefined

mean reversion is undefined

stationary

if autocorrelations are statistically indistinguishable from zero then the time series is said to be _____

1

For an AR(1) time series to be covariance stationary, the absolute value of the b1 coefficient must be less than ____

unit root

If the value of the lag coefficient is equal to one, the time series is said to have a unit root and will follow a random walk process. A series with a unit root is not covariance stationary.

all random walks have a _____ because by definition B1=1

all random walks have a _____ because by definition B1=1

unit root

if a time series has a _____ it is non stationary and you cannot estimate a linear regression

differencing

is a process we use to transform data with a unit root; it is performed by subtracting one value in the time series from another. 𝑦𝑡= 𝑥 𝑡 − 𝑥𝑡−1 = Et

mean reversion level = 0

becomes covariance stationary

mean reversion level = 0

becomes covariance stationary

Smoothing models

remove short-term fluctuations by smoothing out a time series.

n period moving average

moving average models

not used often

MA(1) 𝑥𝑡 = ε𝑡 +θε𝑡−1

For an MA(q) model, the first q autocorrelations will be significantly different from 0, and all autocorrelations beyond that will be equal to 0.

MA(1) 𝑥𝑡 = ε𝑡 +θε𝑡−1

For an MA(q) model, the first q autocorrelations will be significantly different from 0, and all autocorrelations beyond that will be equal to 0.

Autoregressive vs Moving average model

-The autocorrelations for an AR model will generally begin as large values and gradually decline.

-The autocorrelations for a MA model will drop dramatically after q lags are reached, identifying both the MA process and its order.

-The autocorrelations for a MA model will drop dramatically after q lags are reached, identifying both the MA process and its order.

seasonality

Time series that show regular patterns of movement within a year across years.

Seasonal lags are most often included as a lagged value one year before the prior value

For quarterly data, the fourth autocorrelation will not be statistically zero if there is quarterly seasonality. For monthly, the 12th, and so on.

To correct for seasonality, we can include an additional lagged term to capture the seasonality.

Xt = B0 + B1Xt-1 + B2Xt-4 +ei

Seasonal lags are most often included as a lagged value one year before the prior value

For quarterly data, the fourth autocorrelation will not be statistically zero if there is quarterly seasonality. For monthly, the 12th, and so on.

To correct for seasonality, we can include an additional lagged term to capture the seasonality.

Xt = B0 + B1Xt-1 + B2Xt-4 +ei