11. Time Series Analysis Outline

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11. Time Series Analysis

Basic Regression

Read Wooldridge (2013), Chapter 10

11. Time Series Analysis . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat

Outline

I. The Nature of Time Series Data II. Examples of Time Series Models III. Finite Sample Properties of OLS

IV. Functional Form, Dummy Variables, and Index Numbers

I. Nature II. Examples III. Properties IV. Index 2

I. The Nature of Time Series Data

• Time series

A time series data set consists of observations on a variable or several variables over time.

eg. GDP, money supply, interest rate.

• Cross section series

A cross sectional data set is a data set collected from a population at a given point in time.

I. Nature II. Examples III. Properties IV. Index 3

11. Time Series Analysis . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat I. The Nature of Time Series Data

Differences between Time Series and Cross Section

(1) Ordering

Cross section: Ordering usually is not important.

Time series: A data set comes with a temporal ordering (2) Random sample

Cross section: A random sample is drawn from the population. Each observation is randomly drawn (MLR.2).

Time series: An observation is an outcome of random variables.

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Terms:

• Stochastic – random

• Realization – observation

• Formally, say observation 1960‐2003.

A sequence of random variables indexed by time is called a stochastic process or a time series process.

– At a point in time, we obtain a single realization of the stochastic process. Note that we cannot go back in time.

II. Examples of Time Series Regression

• Static and Finite Distributed Lag Models

• Static Model

y t =  0 +  1 z t + u t t = 1, 2,…,n – A static model is a model that relates y to z using the

same time period.

–  1 : An immediate effect of z on y.

11. Time Series Analysis . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat II. Examples of Time Series Regression

Examples of Time Series Regression

• Example : Phillips Curve inf t =  0 +  1 unem t + u t

inf t : annual inflation rate (%) unem t : unemployment rate (%)

• = 1.423+ .468unem (s.e) (1.72) (.289)

[t] [.828] {1.617}

n=49 R 2 =.053 R 2 bar=.032 – How to define  1 ?

Finite Distributed Lag (FDL) Models:

Effect of one or more variables with a lag on y

• Example: Effect of the growth in money supply on economic growth in Thailand over 1993‐2002 (quarterly data)

ggdp: economic growth (percent) gm1: money supply growth (percent) ggdp

_t

= 

₀

+ 

₀

gm1

_t

+ 

₁

gm1

_t‐1

+ 

₂

gm1

_t‐2

+ u

_t

• Interpret coefficients: let t = 0



₀

: “this quarter” effect of MS growth on “this quarter” ggdp.



₁

: “last quarter” effect of MS growth on “this quarter” ggdp



₂

: “two‐quarters ago” effect of MS growth on “this quarter” ggdp.

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Example: effect of gm1 on ggdp y t =  0 +  0 z t +  1 z t‐1 + 2 z t‐2 + u t

t = .001 + .319gm1 t +.121gm1 t‐1 ‐.126gm1 t‐2 p‐value {.88} {.0005} {.105} {.1359}

n=39, R 2 =.5088 F=12.08 {p‐value=.000014}

Interpretation: Economic Significance.

1) Interpret 2) Interpret 3) Interpret

A General form – a FDL of order two y t =  0 +  0 z t +  1 z t‐1 + 2 z t‐2 + u t

Interpret coefficients:

1)  0 : the immediate change in y due to the one‐unit increase in z at time t

–  0 is called the impact propensity or impact multiplier.

2)  0 +  1 + 2 : the long‐run change in y given a permanent increase in z

–  0 +  1 + 2 is called the long run propensity (LRP) or long‐run multiplier.

A temporal increase in z:  j

y t =  0 +  0 z t +  1 z t‐1 + 2 z t‐2 + u t

“temporal” means lasting only for a time.

• Assumptions:

1) Before time t: z is a constant (c) 2) At time t, z increases one unit to c+1

3) After time t, z reverts back to its previous level, c

• Interpretation:

– 

₀

= y

_t

– y

_t‐1

: immediate change in y due to the one‐unit increase in z at time t

– 

₁

= y

_t+1

– y

_t‐1

: the change in y one period after the increase in z.

– 

₂

= y

_t+2

– y

_t‐1

: the change in y two periods after the increase in z.

I. Nature II. Examples III. Properties IV. Index 11

A permanent increase in z:  0 +  1 + 

₂

y t =  0 +  0 z t +  1 z t‐1 +  2 z t‐2 + u t

• Assumptions:

1) Before time t, z is a constant (c).

2) At time t, z increases one unit permanently to c+1.

• Interpretation: With the permanent increase in z



₀

= y

_t

– y

_t‐1

: immediate change



₀

+

₁

= y

_t+1

– y

_t‐1

: increase in y after one period.



₀

+

₁

+

₂

= y

_t+2

– y

_t‐1

: increase in y after two periods.

• 

₀

+

₁

+

₂

is the LR change in y given a permanent increase in z in the FDL model of order 2.

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Example: effect of gm1 on ggdp y t =  0 +  0 z t +  1 z t‐1 +  2 z t‐2 + u t

t = .001 + .319gm1 t +.121gm1 t‐1 ‐.126gm1 t‐2 p‐value {.88} {.0005} {.105} {.1359}

n=39, R 2 =.5088 F=12.08 {p‐value=.000014}

Interpretation: Economic and Statistical Significance?

– Impact multiplier = ? and Test?

– Long run multiplier = ? and Test?

• Graph: a lag distribution with two nonzero lags.

Dependent Variable: GGDP Method: Least Squares Sample(adjusted): 1993:2 2002:4

Included observations: 39 after adjusting endpoints

Variable Coefficient Std. Error t-Statistic Prob.

C 0.001303 0.006156 0.211701 0.8336

GM1 0.319434 0.082622 3.866224 0.0005

GM1(-1) 0.120663 0.072506 1.664191 0.105

GM1(-2) -0.12604 0.082562 -1.52659 0.1359

R-squared 0.50881 Mean dependent var 0.008038

Adjusted R-squared 0.466708 S.D. dependent var 0.040941

S.E. of regression 0.029898 Akaike info criterion -4.08516

Sum squared resid 0.031286 Schwarz criterion -3.91453

Log likelihood 83.66052 F-statistic 12.08517

Durbin-Watson stat 2.033647 Prob(F-statistic) 0.000014

Effect of Monetary Policy on Economic Growth

I. Nature II. Examples III. Properties IV. Index 14

III. Finite Sample Properties of OLS under classical Assumptions

TS.1: Linear in parameters.

Given stochastic process {(x

_t1

, x

_t2

,…,x

_tk

, y

_t

): t = 1, 2, .., n}

y

_t

= 

₀

+ 

₁

x

_t1

+ … + 

_k

x

_tk

+ u

_t

where {u

_t

: t = 1, 2, …., n} is the sequence of errors.

x

_tj

: t denotes the time period j indicates one of the k variables

 Unbiasedness of OLS: TS.1‐TS.3.

11. Time Series Analysis . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat III. Finite Sample Properties of OLS

TS.2: No perfect collinearity

No independent variable is constant or a perfect linear combination of the others.

TS.3 : Zero Conditional Mean E(u t |X) = 0

where X is an array with n rows and k columns.

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Example: y t =GDP; x t1 =M1 t x t2 =GE t (gov’t spending); n=44, k=2

year y

_t

= gdp x

_t1

= M1 x

_t2

= GE error

1960 gdp

₆₀

M1

₆₀

GE

₆₀

u

₆₀

1961 gdp

₆₁

M1

₆₁

GE

₆₁

u

₆₁

1962 gdp

₆₂

M1

₆₂

GE

₆₂

u

₆₂

.. .. .. .. ..

2001 gdp

₀₁

M1

₀₁

GE

₀₁

u

₀₁

2002 gdp

₀₂

M1

₀₂

GE

₀₂

u

₀₂

2003 gdp

₀₃

M1

₀₃

GE

₀₃

u

₀₃

u

_t

is uncorrelated with each explanatory variable in every time period. We say that x

_tj

are strictly exogenous.

 Theorem 3.1: Unbiasedness of OLS

If TS.1 – TS.3 hold, then the OLS estimators are unbiased, conditional in X, i.e.,

E( ) =  j for j = 1, ..., k

TS. 3' implies that u t is uncorrelated with regressors dated at time t, (TS.3' next chapter)

E(u t |x t1 ,…,x tk ) = E(u t |x) = 0

We say x tj are contemporaneously exogenous.

• Why don’t we assume E(u i |X)=0 or strict exogeneity in the cross sectional analysis?

• Random sampling: u i is automatically independent of the explanatory variables other than i.

year y

_t

= gdp x

_t1

= M1 x

_t2

= GE error

2001 gdp

₀₁

M1

₀₁

GE

₀₁

u

₀₁

Example: Strict Exogenity Assumption Model: y t =  0 +  1 z t + u t

ggdp t =  0 +  1 gm1 t + u t TS. 3 requires that

(1) u t and z t are uncorrelated

(2) u t is also unrelated with past and future values of z;

that is, z have no lagged effect on y.

(3) A subtle point : the changes in error term today cannot cause future changes in z. ( u t  y t  z t+1 )

( u t  ggdp t  gm1 t+1 )

• This rules out feedback from y on future values of z.

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Example: Murder rate equation mrdrte t = 0 + 1 polpc t + u t

mrdrte: murders per 10,000 people polpc: number of police in the force

• Two implications:

(1) TS.3 implies that u t is uncorrelated with polpc in all time period.

Violation: Higher u 0 may lead to larger polpc 1 force.

(2) Explanatory variables that are strictly exogeneous cannot react to what has happened to y in the past.

year=t y t =mrdrte t z t =polpc t u t =error

2000 10,000 100 u 0

2001 …. 150 ..

Violation: high u 0  high mrdrt 0  high polpc 1

? ?

Cases: Agricultural Production, rainfall and labor input

Example : Agricultural Production (y)

Case 1: y=output z=rainfall u t =error

t=0 y 2000 u 2000

t=1 z 2001

Case 2: y=output z=labor input u t =error

t=0 z 2000 u 2000

t=1 z 2001

?

TS.4: Homoskedasticity

VAR(u t |X) = VAR(u t ) =  2 t = 1 ,…, n

• Example: Effect on Treasury bills i3 t =  0 +  1 inf t +  2 def t + u t

def t : federal deficit as a percentage of GDP.

• TS. 4 requires that unobservables affecting interest rates have a constant variance over time.

• Violation: Variability of interest rates depends on the level of inflation or relative size of the deficits.

TS.5: No Serial Correlation

Corr(u s ,u t |X) = 0 for all t  s

Example: No Serial Correlation

y t =  0 +  1 inf t +  2 def t + u t

• When u t‐1 > 0 then, on average, the error in the next period u t is positive, or Corr(u t , u t‐1 ) > 0.

– This problem is called serial correlation or auto correlation.

• This implies that if interest rate is high in this period, it will be high in the next period: a violation of TS.5.

I. Nature II. Examples III. Properties IV. Index 24

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25

Positive Serial Correlation

Negative Serial Correlation

Serial Correlation

• Theorem 10.2: OLS Sampling Variances If TS.1 – TS.5 hold, then

Var( X ) =

j = 1, …, k, where SST j is the total sum of squares of x tj and R j 2 is the regression of x j on the other regressors.

• Theorem 10.4: Gauss Markov Theorem

If TS.1‐TS.5 (Gauss Markov assumptions) hold, then conditional on X, the OLS estimators are the best linear unbiased estimators (BLUE).

I. Nature II. Examples III. Properties IV. Index 27

• Theorem 10.3: Unbiased estimator of  2

If TS.1 – TS.5 hold, then the unbiased estimator of  2 is

2 =

Inference under the CLM Assumptions

• Assumption TS.6: Normality assumption The errors u t are independent of X and

u t  N(0, 2 )

That is, u t is independently and identically distributed (i.i.d) as Normal (0, 2 ).

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Inference under the CLM Assumptions

 Theorem 10.5: Normal Sampling Distribution If TS.1‐TS.6 (CLM assumptions) hold, then (1) is normally distributed,

(2) t and F statistics have t and F distributions, respectively, under the null hypothesis, and

(3) the usual construction of confidence interval is valid.

Example: Effects of inflation and deficits on interest rate in the United States over 1948‐66.

i3: the three‐month T‐bill rate

def: federal deficit as a percentage of GDP 3 t = 1.25 + 0.613inf t + 0.700def t

s.e. (0.44) (0.076) (0.118) t [2.84] [8.06] [5.93]

n = 49 R 2 = 0.697 R 2 bar= 0.683

• Interpret: Economic and Statistical Significance

1) Test inf and def at the 5% level.

2) Interpret the coefficient on inf

3) Interpret the coefficient on def

IV. Functional Form, Dummy Variables, and Index number

• Index number: GDP Example

• Base period: 1988

• Base value: 100

1988 1997 1998 1999

Nominal GDP 1,559.8 4,740.2 4,628.4 4,615.4 Real GDP 1,559.8 3,074.5 2743.4 2859.2 Index: GDP Deflator 100.0 154.2 168.7 161.4

11. Time Series Analysis . Quantitative Methods of Economic Analysis . 2949605 . Chairat Aemkulwat IV. Functional Form, Dummy and Index

Change the base year

• NESDB uses 1988 as the base year for GNP deflator 1988 100 (base year)

1997 154

1999 161

• Suppose we want to change the base year to 1997

1988 65 (100/154)*100

1997 100 (base year) (154/154)*100

1999 105 (161/154)*100

100 index *

index index old

base new

t

t 



 



  new old

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Old and new GDP deflator Indexes compared.

1988 1997 1998 1999

Old GDP deflator Index 100.0 154.2 168.7 161.4

New GDP deflator Index 65 100 109 105

Calculate the growth rate of GDP deflator: inflation

100

* year )

year year

( new rate

growth

t t t

old

 old



Calculate the growth rate of GDP deflator: another index for inflation

1988 1997 1998 1999

Old GDP deflator Index 100.0 154.2 168.7 161.4 New GDP deflator Index 64.9 100.0 109.4 104.7

Inflation rate 54.2 9.4 -4.3

Inflation in 1999 = [(104.7‐109.4)/109.4]*100

= ‐4.3%

Example : Effect of Personal Exemption on Fertility Rates;

gfr t =  0 +  1 pe t +  2 ww2 t +  3 pill t + u t

gfr t : general fertility rate – the number of children born to every 1,000 women.

pe t : real dollar value of the personal tax exemption ww2 = 1 during WW II (1941‐1945)

= 0 otherwise (1913‐40,1946‐84) pill = 1 from 1963 on (1963‐84)

= 0 otherwise (1913‐62)

Estimation:

t

= 98.68 + .083pe

_t

– 24.24ww2

_t

– 31.59pill

_t

s.e. (3.21) (.030) (7.46) (4.08)

t {30.74} {2.77} {3.25} {7.74}

n = 72 R

²

= 0.473 R

²

bar = 0.450 What can you say about statistical significance?

Dependent Variable: GFR Method: Least Squares Sample: 1 72

Included observations: 72

C 98.68176 3.208129 30.75991 0

PE 0.08254 0.029646 2.784166 0.0069

WW2 -24.2384 7.458253 -3.249876 0.0018

PILL -31.594 4.081068 -7.74161 0

Adjusted R-squared 0.450184 S.D. dependent var 19.80464

S.E. of regression 14.68506 Akaike info criterion 8.265492

Sum squared resid 14664.27 Schwarz criterion 8.391973

Log likelihood -293.558 F-statistic 20.37801

Durbin-Watson stat 0.176873 Prob(F-statistic) 0

Fertility Rate Equation with no lags

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Example: fertility rate equation with a lag

t

= 95.87 + .073pe

_t

‐ .0058pe

_t‐1

+ .034pe

_t‐2

‐ 22.31ww2

_t

‐ 31.30pill t‐stat {29.07} {.579} {‐.030} {.269} {11.58} {7.86}

n =70 R

²

= 0.499 R

²

‐bar= 0.459

• Interpretation:

1) Interpret the coefficient on ww2.

2) Interpret the coefficient on pill.

3) Are pe t , pe t‐1 , pe t‐2 individually statistically significant?

4) Are pe t , pe t‐1 , pe t‐2 jointly statistically significant?

– Problem: multicollinearity??

Dependent Variable: GFR Sample(adjusted): 3 72

Included observations: 70 after adjusting endpoints

C 95.8705 3.281957 29.21138 0

PE 0.072672 0.125533 0.578906 0.5647

PE(-1) -0.00578 0.155663 -0.037129 0.9705

PE(-2) 0.033827 0.126257 0.267919 0.7896

WW2 -22.1265 10.73197 -2.061737 0.0433

PILL -31.305 3.981559 -7.862495 0

Adjusted R-squared 0.459427 S.D. dependent var 19.40881 S.E. of regression 14.27008 Akaike info criterion 8.236023 Sum squared resid 13032.64 Schwarz criterion 8.428751

Wald Test:

Equation: Untitled

Test Statistic Value df Probability

F-statistic 0.05343 (2, 64) 0.948

Chi-square 0.10686 2 0.948

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

C(3) -0.00578 0.155663

C(4) 0.033827 0.126257

Restrictions are linear in coefficients.

View/Coefficient Tests/Wald-Coefficient Restrictions

Redundant Variables: PE PE(-1) PE(-2)

F-statistic 3.972964 Probability 0.011652 Log likelihood ratio 11.95477 Probability 0.00754

View/Coefficient Tests/Redundant Variables…

Long run propensity...

gfr

_t

= 

₀

+ 

₀

pe

_t

+ 

₁

pe

_t‐1

+ 

₂

pe

_t‐2

+ 

₂

ww2

_t

+ 

₃

pill

_t

+ u

_t

t

= 95.87 + .073pe

_t

‐ .0058pe

_t‐1

+ .034pe

_t‐2

‐ 22.31ww2

_t

‐ 31.30pill t {29.07} {.579} {‐.030} {.269} {11.58} {7.86}

5) Should we ditch pe t‐1 or pe t‐2 ?

6) What is the long‐run propensity in this model?

7) Is the long‐run propensity statistically significant?

What is the null hypothesis?

8) Find a 95% confidence interval of the long‐run propensity.

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Dependent Variable: GFR Sample(adjusted): 3 72

Included observations: 70 after adjusting endpoints

C 95.8705 3.281957 29.21138 0

PE 0.100719 0.029803 3.379532 0.0012

PE(-1)-PE -0.00578 0.155663 -0.037129 0.9705

PE(-2)-PE 0.033827 0.126257 0.267919 0.7896

WW2 -22.1265 10.73197 -2.061737 0.0433

PILL -31.305 3.981559 -7.862495 0

Adjusted R-squared 0.459427 S.D. dependent var 19.40881 S.E. of regression 14.27008 Akaike info criterion 8.236023 Sum squared resid 13032.64 Schwarz criterion 8.428751

The 95% CI of LRP

• The 95% confidence interval of LRP is

Given = 0.101 and s.e.( ) = 0.030, the 95% CI is

 c*s.e.( ) = .101  2(.030)

= .041 to .160

• c: is the 95 th percentile in the t distribution with 64 DF n = 70 k = 5

n – k –1 = 70 –5 – 1= 64 c = 2.00 (DF = 60)

= 1.987 (DF = 90)

Recap of Time Series Analysis

• The Nature of Time Series Data

• Examples of Time Series Models

• Finite Sample Properties of OLS

• Functional Form, Dummy Variables, and Index Numbers