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1. Some basics of time-series
and forecasting
2019 Copyright QuantUniversity LLC.
Presented By:
Gustavo Vicentini,PhD
QuantUniversity
Econometrics Lead

2. • Gustavo leads Econometrics and Forecasting
programs at QuantUniversity.
• Gustavo worked several years as an economist at
Analysis Group and currently serves as
an associate teaching professor in the Department
of Economics at Northeastern University.
• Gustavo received a PhD in economics from Boston
University.
Gustavo Vicentini, PhD
Econometrics Lead
2

3. Principles of Econometrics, 4th Edition Page 3
Chapter 9: Regression with Time Series Data:
Stationary Variables
Some basics of time-series
and forecasting
Gustavo Vicentini
Quant University – Econometrics Lead

4. Principles of Econometrics, 4th Edition Page 4
Chapter 9: Regression with Time Series Data:
Stationary Variables
Introduction (I)
U.S. real GDP growth (Qtr)
Time-series data are widely encountered in macroeconomics, finance, and business
U.S. real GDP (Qtr, millions) 3-year U.S. bond rate Australian inflation rate
U.S. retail auto sales (month) Price of Bitcoin and of Gold

5. Principles of Econometrics, 4th Edition Page 5
Chapter 9: Regression with Time Series Data:
Stationary Variables
¡ Time-series Econometrics is very useful to analyze time-
series data
¡ Policy analysis
¡ Quantification of monetary policy: If Fed lowers
interest rate this quarter, how is current and future
GDP growth affected?
¡ Forecasting
¡ Using past data on inflation, what is a good
forecast for future inflation?
¡ Before using regression on time-series data, one needs to
test whether the series are stationary or non-stationary
¡ Why? Non-stationary series can give spurious
regression results if not handled properly
Introduction (II)

6. Principles of Econometrics, 4th Edition Page 6
Chapter 9: Regression with Time Series Data:
Stationary Variables
Modelling stationarity vs non-stationarity (I)
A series is non-stationary if its mean and/or variance are
not constant over time. It either trends (upward or
downward) or it wanders around with slow turns without
necessarily returning to its mean
A series is stationary if its
mean and variance are
constant over time. That is,
the series hovers around its
mean with an equal amount of
variability over time

7. Principles of Econometrics, 4th Edition Page 7
Chapter 9: Regression with Time Series Data:
Stationary Variables
¡ The AR(1) model (“Auto-Regressive of order 1”) is a
useful univariate model for explaining stationarity vs non-
stationarity
yt = α + ρ yt-1 + vt
¡ vt ~ iid with mean zero and constant variance (σ2
v)
¡ If |ρ| < 1, model is stationary:
E(yt) = α / (1 – ρ) (doesn’t depend on t)
Var(yt) = σ2
v / (1 – ρ2) (doesn’t depend on t)
¡ If ρ = 1, model is non-stationary (unit root, random walk):
E(yt) = t·α + y0 (does depend on t)
Var(yt) = t·σ2
v (does depend on t)
Modelling stationarity vs non-stationarity (II)

8. Principles of Econometrics, 4th Edition Page 8
Chapter 9: Regression with Time Series Data:
Stationary Variables
¡ Even more general AR(1) model: yt = α + ρ yt-1 + λ t + vt
Modelling stationarity vs non-stationarity (III)
yt = 0.7 yt-1 + vt yt = 1 + 0.7 yt-1 + vt yt = 1 + 0.7 yt-1 + 0.01t + vt
Unit root
Stationary with mean > 0 “Stationary” around a trend
yt = 1· yt-1 + vt
Stationary with mean = 0
yt = 0.1 + 1· yt-1 + vt
Unit root with drift
yt = 0.1 + 1· yt-1 + 0.01t + vt
Unit root with drift and trend

9. Principles of Econometrics, 4th Edition Page 9
Chapter 9: Regression with Time Series Data:
Stationary Variables
¡ Several simulated unit roots: yt = 1· yt-1 + vt ; vt ~ N(0,1)
¡ Note how the variance of the series depends on time
Modelling stationarity vs non-stationarity (IV)

10. Principles of Econometrics, 4th Edition Page 10
Chapter 9: Regression with Time Series Data:
Stationary Variables
¡ To test whether a series is stationary or non-stationary
one needs to test whether:
ρ = 1 (non-stationary) vs ρ < 1 (stationary)
¡ This is the Dickey-Fuller test
¡ If non-stationary (ρ = 1), then convert to its stationary
version before analyzing it. Otherwise spurious regression
¡ If series already stationary (ρ < 1), then it’s ready for
regression
¡ For the time being, let’s assume your data are stationary
and “ready” for policy analysis and forecasting
¡ (We’ll return to Dickey-Fuller later)
Modelling stationarity vs non-stationarity (V)

11. Principles of Econometrics, 4th Edition Page 11
Chapter 9: Regression with Time Series Data:
Stationary Variables
An example of policy analysis:
Monetary policy
Quarterly data on GDP growth (“y” variable) and interest rate (“x”)
How does a change in interest rate now affects GDP growth over time
(i.e., monetary policy)?
Regress y on “q” lagged values of x
yt = β0 + β1 xt-1 + β2 xt-2 + … + βq xt-q + vt
β1 is the one-quarter-ahead effect, β2 is the two-quarters-ahead
effect, and so on
If both y and x stationary, then use OLS regression to estimate the
β’s. This provides a quantification of monetary policy
A lag selection criterion should be used to select “q”
– Popular ones are AIC and BIC (similar to R2)
Can also include “quarter dummies” to control for seasonality
Can also use lagged values of yt to improve fit

12. Principles of Econometrics, 4th Edition Page 12
Chapter 9: Regression with Time Series Data:
Stationary Variables
An application of policy analysis:
Okun’s Law (I)
Okun’s Law: “U” = unemployment rate; “G” = GDP growth
Change in unemployment depends on how growth deviates from
“normal” growth rate “GN”
Its “econometric” version:
Lags of growth can be included:
If both “DU” and “G” series are stationary, then OLS
regression can be used to estimate the β’s
( )1t t t NU U G Gg-- = - -
0βt t tDU G ea= + +
0 1 1 2 2β β β βt t t t q t q tDU G G G G ea - - -= + + + + + +!

13. Principles of Econometrics, 4th Edition Page 13
Chapter 9: Regression with Time Series Data:
Stationary Variables
An application of policy analysis:
Okun’s Law (II)
§ Quarterly U.S. data:
“DU ” “G ”

14. Principles of Econometrics, 4th Edition Page 14
Chapter 9: Regression with Time Series Data:
Stationary Variables
An application of policy analysis:
Okun’s Law (III)
§ OLS regression results:
§ Recall: Number of lags can be selected with AIC or BIC

15. Principles of Econometrics, 4th Edition Page 15
Chapter 9: Regression with Time Series Data:
Stationary Variables
An example of forecasting:
Monetary policy
After estimating the β’s, then forecast future growth. A
one-quarter-ahead (“T+1”) forecast is ŷT+1:
ŷT+1 = β̂0 + β̂1 xT + β̂2 xT-1 + … + β̂q xT-q+1
The β̂ ’s are the OLS estimates
Then calculate a standard error (S.E.) for the forecast ŷT+1,
and use it to construct a 95% confidence interval for ŷT+1:
95% C.I. for ŷT+1 = ( ŷT+1 – 1.96·SE , ŷT+1 + 1.96·SE )
Could also calculate a two-quarters-ahead forecast and
confidence interval, and so on

16. Principles of Econometrics, 4th Edition Page 16
Chapter 9: Regression with Time Series Data:
Stationary Variables
An application of forecasting:
Future growth based on past growth (I)
Consider an AR(2) model for real GDP growth:
Gt = β0 + β1 Gt-1 + β2 Gt-2 + vt
Using data from 1986Q1 to 2009Q3, OLS estimates yield:
β̂0 = 0.467 β̂1 = 0.377 β̂2 = 0.246
Given that:
GT = G2009Q3 = 0.8% and GT-1 = G2009Q2 = -0.2%,
The one-quarter-ahead forecast is:
Ĝ2009Q4 = ĜT+1 = β̂0 + β̂1 GT + β̂2 GT-1
Ĝ2009Q4 = ĜT+1 = 0.718%

17. Principles of Econometrics, 4th Edition Page 17
Chapter 9: Regression with Time Series Data:
Stationary Variables
An application of forecasting:
Future growth based on past growth (II)
The two-quarters-ahead forecast is:
Ĝ2010Q1 = ĜT+2 = β̂0 + β̂1 ĜT+1 + β̂2 GT = 0.933%
And so on into further quarters…
After computing the standard errors, we can tabulate the
forecasts and confidence intervals:

18. Principles of Econometrics, 4th Edition Page 18
Chapter 9: Regression with Time Series Data:
Stationary Variables
Many economic/finance data are non-stationary (unit root):
If series is non-stationary, it shouldn’t be used (in its “raw” form) in a
regression, because OLS estimator will not behave well
Need to transform into a stationary series and then use that in the
regression
The transformation is just its first-difference: Dyt = yt – yt-1
For example, if both y and x are non-stationary, you should
Use: Dyt = β0 + β1 Dxt + ut And not use: yt = β0 + β1 xt + ut
Handling non-stationary series (I)
3-year U.S. bond rate Bitcoin price U.S. inflation rate

19. Principles of Econometrics, 4th Edition Page 19
Chapter 9: Regression with Time Series Data:
Stationary Variables
Examples of the first-difference transformation:
Handling non-stationary series (II)

20. Principles of Econometrics, 4th Edition Page 20
Chapter 9: Regression with Time Series Data:
Stationary Variables
§ Why can’t use “raw” version of non-stationary series in a regression?
Because the potential of “spurious regression”
§ Consider two unit roots that were independently simulated (they are not
related to each other)
§ An OLS regression of rw1 on rw2 yields:
§ These results are statistically significant (t = 40.8 and R2 = 0.70!!), but
completely meaningless, or spurious. The apparent significance is false
Handling non-stationary series (III)
2
1 217.818 0.842 , 0.70
( ) (40.837)
t trw rw R
t
= + =

21. Principles of Econometrics, 4th Edition Page 21
Chapter 9: Regression with Time Series Data:
Stationary Variables
First thing is to test whether the series is non-stationary
If series is stationary, then work with the series itself
(without taking first-difference)
If series is non-stationary, then work with first-difference
Dickey-Fuller (“DF”) test used to test non-stationarity
Consider again the AR(1) model:
A more convenient form is:
Run this last regression, and test whether γ = 0 or γ < 0
Handling non-stationary series (IV)
1t t ty y v-= r +
( )
1 1 1
1
1
1
t t t t t
t t t
t t
y y y y v
y y v
y v
- - -
-
-
- = r - +
D = r - +
= g +

22. Principles of Econometrics, 4th Edition Page 22
Chapter 9: Regression with Time Series Data:
Stationary Variables
Dickey-Fuller critical values for γ̂
If γ̂ is less then the critical value(s), we reject the
hypothesis of non-stationarity
Handling non-stationary series (V)
1t t ty y v-= r + ( )
1 1 1
1
1
1
t t t t t
t t t
t t
y y y y v
y y v
y v
- - -
-
-
- = r - +
D = r - +
= g +