Study Session 3 - Time Series Analysis Flashcards Preview

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Flashcards in Study Session 3 - Time Series Analysis Deck (25)
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1
Q

What is a time series? What is a trend? Types of Trend?

A

a set of observations for a variable over successive periods of time. Has a trend if you can see a pattern can be seen by plotting the data.

Only difference between this equation and simple regression is that Yt becomes data point from the time series to predict the next.

Linear or Log
Log convex - growth, concave - decay

2
Q

How is the log exponential transformed to linear?

A

ln(yt)= ln(e^b₀+b₁(t)= ln(yt) = b₀+b₁(t) - can now be used in linear regression.

3
Q

When do you use a linear trend model? Log-linear model?

A

When is appears the data points are equally above and below the regression line, use linear. If curved or persistently above or below the regression line then you log-linear. If it is growing at a constant rate use log-linear.

4
Q

What do you use when log-linear is appropriate but exhibits serial correlation?

A

Autoregressive.

5
Q

What is an autoregressive model?

A

When the dependent variable is regressed against one or more lagged values (previous periods) of itself.
Must be covariance stationary or inferences based on the model are not valid.

6
Q

What are the three conditions for covariance stationary?

A

The expected value of the time series in constant over time. (mean reverting level).

The time series volatility around its mean doesn’t change over time.

The covariance of the time series with leading or lagged values of itself is constant.

Constant EXPECTED VALUE, VARIANCE, and COVARIANCE.

7
Q

What is the chain rule of forecasting?

A

Because you are working with lagged values, you must calculate one step forward before two step forward, etc.

8
Q

How do you test for serial correlation in an autoregressive model?

A
  • Calculate the autocorrelations of the model’s residuals (the level of correlation between the errors - forecast v. actual, one period to the next)
  • Test whether the autocorrelations are significantly different from zero using a t test where

t= correlation between 𝝴t, 𝝴t-k/ 1/√T

correlation between error term t and kth lagged error term.
df = T-2

9
Q

What is mean reversion?

A

time series exhibits mean reversion if it has a tendency to move towards it’s mean. If we are at the mean reverting level, the next predicted value will be the same.

So if xt > larger than mean reverting level, next will be lower than xt and vice versa.

10
Q

What is the formula for mean reversion?

A

xt= b₀/(1-b₁)

11
Q

Difference between in-sample forecasts and out of sample forecasts?

A

In sample are within the range of the data (time period)

out of sample are made outside the range. Tests whether the model adequately describes the time series.

12
Q

What is the root mean squared error criterion (RMSE)?

A

used to compare the accuracy of autoregressive models in forecasting out of sample values. Lower, better forecast.

13
Q

What is instability of variables?

A

Also known as none stationary. Data can be dynamic, variables changing all the time.

14
Q

What is a random walk? With constant drift?

A

the predicted value of the series in one period is equal to the value of a series in the previous period plus a random error term.

xt = xt-1 +𝝴t where the best forecast is t, t-1 and the expected value of each error term is zero, the variance is constant, and there is no serial correlation.

Constant drift increases or decreases the walk by the same amount each period. It’s an intercept. +/-

15
Q

Why is a random walk not covariance stationary?

A

Must have a finite mean-reverting value.

xt = b0/0, not defined.

16
Q

What is a unit root?

A

b₁ = 1

17
Q

What is the Dickey Fuller test?

A

Tests whether a time series is covariance stationary.

Use t-test to test if it is different.

Expressed at b₁-1=g

test is H₀: g=0

18
Q

How do we transform a random walk time series to covariance stationary?

A

Using first differencing.

19
Q

Describe first differencing.

A

Subtracting the values of the time series in the immediately preceding period from the current. (t-(t-1)) to define a new dependent variable, y.

yt = xt - xt-1 –> yt = 𝝴t

As an AR model: yt = b0 + b1yt-1 + et

b0 = b1 = 0

so, 0/1-0 = 0, so finite and covariance stationary.

20
Q

How to tell if there is seasonality in the regression? How to correct it?

A

Look for the t test from the independent variable that can be rejected.

Correct by adding in an additional lag from same period last year as a independent variable.

21
Q

What is ARCH?

A

AutoRegressive Conditional Heteroskedasticity. Means the variance of the residuals in one period is dependent on the variance of the residual in a previous period.

Causes the standard errors of each coefficient to be wrong, hypothesis test invalid.

22
Q

How do you test dual time series (xt+yt) for non stationary?

A

Run DF tests with five possible results:
Both are good - go ahead.
Only dependent variable time series is covariance stationary
Only independent variable time series is covariance stationary
Neither is covariance stationary and are not cointegrated.
Neither is covariance stationary but are cointegrated.

23
Q

What is cointegration?

A

Means that two time series are economically linked and follow the same trend that is not expected to change. If they are cointegrated shouldn’t be any problems during analysis.

24
Q

How do you determine what time series model to use?

A
  1. Determine your goal - Trend vs. Cointegrated time series
  2. Plot the data and look for non stationary, heterosk., non constant mean, seasonality, etc.
  3. No seasonality or structure - use trend.
  4. Run analysis, Durbin Watson for Serial corr. if no, use, otherwise AR model instead.
  5. Check for stationarity - if linear, first difference the data. If exponential, first diff the natural log. If structural, run two separate models., if seasonal, incorporate into model.
  6. Run AR, check again for serial. If none, use. If still serial, incorporate lagged value until it’s gone.
  7. Test for ARCH
  8. Check quality by calculating RMSE.
25
Q

What is a structural change?

A

Problem with a time-series model where there is a significant shift in the plotted data at a point in that seems to divide the data in two.