TOPIC 6: BASIC REGRESSION ANALYSIS
WITH TIME SERIES DATA
Time Series Data: What’s Different?
Lags, Differences, Autocorrelation
Dynamic Causal Effects
Classical Assumptions
1
Trend and seasonality
TIME SERIES DATA: WHAT’S DIFFERENT?
Time series data are data collected on the same
observational unit at multiple time periods
 Aggregate consumption and GDP for a country (for
example, 20 years of quarterly observations = 80
observations)
 Yen/$, pound/$ and Euro/$ exchange rates (daily data
for 1 year = 365 observations)
 Cigarette consumption per capita in California, by year
(annual data)
2
TIME SERIES DATA: WHAT’S DIFFERENT?


Answer quantitative questions for which cross-sectional data
are inadequate.
dynamic causal effect: what is the causal effect on a variable
of interest, Y, of a change in another variable, X, over time?
 If the Fed increases the Federal Funds rate now, what will
be the effect on the rates of inflation and unemployment in
3 months? in 12 months?

What is the effect over time on cigarette consumption of a
hike in the cigarette tax?
3
TIME SERIES DATA: WHAT’S DIFFERENT?
economic forecasting: what is your best forecast of the value
of some variable at a future date?
 E.g., what is your best forecast of next month’s unemployment
rate, interest rates, or stock prices?

•
Modeling risks used in financial markets (e.g., modeling
changing variances and “volatility clustering”)
•
Applications outside of economics include environmental and
climate modeling, engineering (system dynamics), computer
science (network dynamics),…

Time series data pose special challenges, and overcoming
those challenges requires some new techniques.
4
TIME SERIES DATA: WHAT’S DIFFERENT?

Unlike cross-section data, time series data has a temporal
ordering: the past can affect the future, but not vice versa.

Need to alter some of our assumptions since we no longer have
a random sample of individuals.

Instead, we have a stochastic (i.e. random) process, i.e. a
sequence of random variables indexed by time. One
realization of a stochastic (i.e. random) process is a time series
data.
5
TIME SERIES DATA RAISES NEW
TECHNICAL ISSUES
Time lags
• Correlation over time (serial correlation, a.k.a.
autocorrelation – which we encountered in panel data)
• Calculation of standard errors when the errors are
serially correlated
A good way to learn about time series data is to
investigate it yourself! A great source for U.S. macro
time series data, and some international data, is the
Federal Reserve Bank of St. Louis’s FRED database.
 The following are some U.S. macro and financial time
series, from FRED
6
 https://fred.stlouisfed.org/series/GDPC1
•
LOADING DATA IN STATA FROM FRED
First install the freduse utility.
. ssc install freduse
 If you know the name of the series you want, you can then
directly import them into STATA using the freduse
command.
 For example, quarterly real GDP is GDPC1, and quarterly
real GDP percent change from previous period is
A191RL1Q225SBEA.
. freduse GDPC1 A191RL1Q225SBEA
 The exact spelling, including capitalization, is required.
 If the command is successful, the variables are now in the
STATA file, with these names.
 Also two time indices are included, date and daten. The
date formats are off
 creating a time index.

7
LOADING DATA IN STATA FROM FRED
create a time index using the tsmktim
. ssc install tsmktim

Suppose the first observation is the 1st quarter of 1947.
. generate time=tq(1947q1)+_n-1
. format time %tq
. tsset time
 The format command formats the variable time with the
time-series quarterly format. The “tq” refers to “time series
quarterly”.
 The tsset command declares that the variable time is the
time index.
 Alternatively
. tsmktim time, start(1947q1)
. tsset time

8
LOADING DATA IN STATA FROM FRED


To learn the desired name of the series you want (you want a
precise series, so do not guess!), go to FRED website.
After loading the data into STATA, you should verify that the
data loaded correctly. Examine the entries using the Data
Editor. Make a time series plot. Also, change the variable
names into something more convenient for use.
9
U.S. REAL GDP
U.S. REAL GDP (PERCENTAGE CHANGE)
SOME MONTHLY U.S. MACRO AND
FINANCIAL TIME SERIES
12
SOME MONTHLY U.S. MACRO AND
FINANCIAL TIME SERIES
13
SOME MONTHLY U.S. MACRO AND
FINANCIAL TIME SERIES
14
A DAILY U.S. FINANCIAL TIME SERIES:
15
TIME SERIES BASICS



Notation
Lags, first differences, and growth rates, and first log
diference approximation to growth rates
Autocorrelation (serial correlation)
16
NOTATION



Yt = value of Y in period t.
Data set: {Y1,…,YT} are T observations on the time
series variable Y
We consider only consecutive, evenly-spaced
observations (for example, monthly, 1960 to 1999, no
missing months) (missing and unevenly spaced data
introduce technical complications)
17
LAGS, FIRST DIFFERENCES, AND GROWTH
RATES
Lags, First Differences, Logarithms, and
Growth Rates
 The first lag of a time series Yt is Yt–1; its jth lag is
Yt–j.
 The first difference of a series, ΔYt, is its change
between periods t – 1 and t, that is, ΔYt = Yt – Yt–1.
 The first difference of the logarithm of Yt is Δln(Yt)
= ln(Yt) – ln(Yt–1).
 The percentage change of a time series Yt between
periods t – 1 and t is approximately 100Δln(Yt),
where the approximation is most accurate when
the percentage change is small.
18
STATA COMMANDS FOR SOME TIME SERIES
OPERATORS
19
EXAMPLE: QUARTERLY RATE OF GROWTH OF
U.S. GDP AT AN ANNUAL RATE
GDP = Real GDP in the US (Billions of $2009)

GDP in the fourth quarter of 2016 (2016:Q4) = 16851

GDP in the first quarter of 2017 (2017:Q1) = 16903

Percentage change in GDP, 2016:Q4 to 2017:Q1
 16903 − 16851 
=
100 × 
0.31%
=
16851




Percentage change in GDP, 2012:Q1 to 2012:Q2, at
an annual rate = 4 × 0.31% = 1.23% ≈ 1.2% (percent
per year)
Using the logarithmic approximation to percent
changes yields 4 × 100 × [log(16903) – log(16851)] =
1.232%
20
EXAMPLE: GDP AND ITS RATE OF CHANGE
21
AUTOCORRELATION (SERIAL
CORRELATION)
The correlation of a series with its own lagged values is
called autocorrelation or serial correlation.
 The first autocovariance of Yt is cov(Yt,Yt–1)
 The first autocorrelation of Yt is corr(Yt,Yt–1)
 Thus
=
corr(Yt , Yt −1 )

cov(Yt , Yt −1 )
= ρ1
var(Yt ) var(Yt −1 )
These are population correlations – they describe the
22
population joint distribution of (Yt, Yt–1)
AUTOCORRELATION (SERIAL
CORRELATION) AND AUTOCOVARIANCE
The jth autocovariance of a series Yt is the covariance
between Yt and its jth lag, Yt–j, and the jth autocorrelation
coefficient is the correlation between Yt and Yt – j. That is,
j th autocovariance = cov(Yt , Yt − j )
j autocorrelation
= ρ=
corr(Yt , Yt − =
j
j)
th
cov(Yt , Yt − j )
var(Yt ) var(Yt − j )
(14.3)
.
(14.4)
The jth autocorrelation coefficient is sometimes called the
jth serial correlation coefficient.
23
SAMPLE AUTOCORRELATIONS
The jth sample autocorrelation is an estimate of the
jth population autocorrelation:
where
cov(�
𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡−𝑗𝑗 )
𝜌𝜌�𝑗𝑗 =
�
var(
𝑌𝑌𝑡𝑡 )
𝑇𝑇
1
�
cov( 𝑌𝑌𝑡𝑡 , 𝑌𝑌𝑡𝑡−𝑗𝑗 ) = � (𝑌𝑌𝑡𝑡 − 𝑌𝑌̄𝑗𝑗+1,𝑇𝑇 ) (𝑌𝑌𝑡𝑡−𝑗𝑗 − 𝑌𝑌̄1,𝑇𝑇−𝑗𝑗 )
𝑇𝑇
𝑡𝑡=𝑗𝑗+1
where Y j +1,T is the sample average of Yt computed over
observations t= j + 1, , T . NOTE :
the summation is over t=j+1 to T
 The divisor is T, not T – j (this is the conventional
definition used for time series data)

24
EXAMPLE
Chinese real GDP, yearly, 1978-2013
 Stata commands
 use gdp_china1.dta,clear
 tsset year
 *declare data to be time series

tsline y,xlabel(1980(5)2010)
 tsline lny,xlabel(1980(5)2010)
 tsline dlny,xlabel(1980(5)2010)
 *tsline draws line plots for time-series data.



gen grow=(y-l.y)/l.y
tsline dlny grow,xlabel(1980(5)2010) lpattern("." "--")
25
0
20000
40000
gdp
60000
80000
100000
TIME SERIES PLOT OF GDP
1980
1985
1990
1995
Year
2000
2005
2010
26
8
9
lngdp
10
11
12
TIME SERIES PLOT OF LN(GDP)
1980
1985
1990
1995
Year
2000
2005
2010
27
.04
.06
dlngdp
.08
.1
.12
.14
TIME SERIES PLOT OF DLN(GDP)
28
1980
1985
1990
1995
Year
2000
2005
2010
LOGARITHMIC APPROXIMATION TO
.05
.1
.15
PERCENT CHANGES
1980
1985
1990
1995
Year
dlngdp
2000
grow
2005
2010
29
AUTOCORRELATIONS
A correlogram, also know n as Auto Correlation Function
(ACF) plot, is a graphic w ay to demonstrate serial
correlation in data
30
AUTOCORRELATIONS
31
AUTOCORRELATIONS
32
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Example: US inflation and unemployment rates
1948-2003
Time series analysis focuses on modeling the
dependency of a variable on its own past, and
on the present and past values of other
variables.
33
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA
Examples of time series regression models
 Static models



In static time series models, the current value of one
variable is modeled as the result of the current
values of explanatory variables
Examples for static models
There is a contemporaneous relationship between
unemployment and inflation (= Phillips curve).
The current murder rate is determined by the current conviction rate, unemployment
rate, and the fraction of young males in the population.
34
DYNAMIC CAUSAL EFFECTS AND THE
DISTRIBUTED LAG MODEL

Dynamic effects necessarily occur over time. The
econometric model used to estimate dynamic causal
effects needs to incorporate lags.
The distributed lag model is:
yt =𝛼𝛼 0 +𝛿𝛿 0 zt +𝛿𝛿 1 zt-1 + … +𝛿𝛿 r zt–r + ut



𝛿𝛿 0 = impact effect of change in z = effect of change
in zt on yt, holding past zt constant
𝛿𝛿 1 = 1-period dynamic multiplier = effect of change
in zt–1 on yt, holding constant zt, zt–2, zt–3,…
𝛿𝛿 2 = 2-period dynamic multiplier (etc.) = effect of
change in zt–2 on yt, holding constant zt, zt–1, zt–3,…
35
DISTRIBUTED LAG MODELS

Cumulative dynamic multipliers


The 2-period cumulative dynamic multiplier is 𝛿𝛿 0 +𝛿𝛿 1
+𝛿𝛿 2 = impact effect + 1-period effect + 2-period effect
regress gfr l(0/2).pe, r
*Stata command; l(0/2).pe:
 Example

pet , pet-1 , pet-2
The fertility rate may depend on the tax value of a
child, but for biological and behavioral reasons, the
effect may have a lag
Children born per
1,000 women in year t
Tax
exemption in
year t
Tax
exemption in
year t - 1
Tax exemption
in year t - 2
36
Distributed Lag models
yt =α 0 + δ 0 zt + δ1 zt −1 + δ 2 zt − 2 + ut

How does one unit of a temporary change in z affect y?

How does one unit of a permanent change in z affect y?
37
Distributed Lag models

A temporary change: Impact propensity, dynamic multiplier
𝛼𝛼0 + 𝛿𝛿0 𝑐𝑐 + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 𝑐𝑐,
𝑠𝑠 < 𝑡𝑡;
𝛼𝛼0 + 𝛿𝛿0 (𝑐𝑐 + 1) + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 𝑐𝑐, 𝑠𝑠 = 𝑡𝑡;
𝑐𝑐,
𝑠𝑠 < 𝑡𝑡;
𝑧𝑧𝑠𝑠 = �𝑐𝑐 + 1, 𝑠𝑠 = 𝑡𝑡; ⇒ 𝑦𝑦𝑠𝑠 = 𝛼𝛼0 + 𝛿𝛿0 𝑐𝑐 + 𝛿𝛿1 (𝑐𝑐 + 1) + 𝛿𝛿2 𝑐𝑐, 𝑠𝑠 = 𝑡𝑡 + 1; ⇒ 𝑦𝑦𝑠𝑠 − 𝑦𝑦𝑡𝑡−1 =
𝑐𝑐,
𝑠𝑠 > 𝑡𝑡;
𝛼𝛼0 + 𝛿𝛿0 𝑐𝑐 + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 (𝑐𝑐 + 1), 𝑠𝑠 = 𝑡𝑡 + 2;
𝛼𝛼0 + 𝛿𝛿0 𝑐𝑐 + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 𝑐𝑐,
𝑠𝑠 > 𝑡𝑡 + 2;

? 𝑠𝑠 = 𝑡𝑡;
? 𝑠𝑠 = 𝑡𝑡 + 1;
? 𝑠𝑠 = 𝑡𝑡 + 2;
? 𝑠𝑠 > 𝑡𝑡 + 2.
A permanent change: cumulative dynamic multipliers and long-run
propensity (LRP) or multiplier
𝛼𝛼0 + 𝛿𝛿0 𝑐𝑐 + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 𝑐𝑐,
𝑠𝑠 < 𝑡𝑡;
? 𝑠𝑠 = 𝑡𝑡;
𝑠𝑠 = 𝑡𝑡;
𝛼𝛼0 + 𝛿𝛿0 (𝑐𝑐 + 1) + 𝛿𝛿1 𝑐𝑐 + 𝛿𝛿2 𝑐𝑐,
𝑐𝑐,
𝑠𝑠 < 𝑡𝑡;
𝑧𝑧𝑠𝑠 = �
⇒ 𝑦𝑦𝑠𝑠 =
⇒ 𝑦𝑦𝑠𝑠 − 𝑦𝑦𝑡𝑡−1 = �? 𝑠𝑠 = 𝑡𝑡 + 1;
𝑐𝑐 + 1, 𝑠𝑠 ≥ 𝑡𝑡;
𝛼𝛼0 + 𝛿𝛿0 (𝑐𝑐 + 1) + 𝛿𝛿1 (𝑐𝑐 + 1) + 𝛿𝛿2 𝑐𝑐,
𝑠𝑠 = 𝑡𝑡 + 1;
? 𝑠𝑠 ≥ 𝑡𝑡 + 2.
𝛼𝛼0 + 𝛿𝛿0 (𝑐𝑐 + 1) + 𝛿𝛿1 (𝑐𝑐 + 1) + 𝛿𝛿2 (𝑐𝑐 + 1), 𝑠𝑠 ≥ 𝑡𝑡 + 2.

LRP: long run change in y given a permanent increase in z.
38
Distributed Lag models
Suppose yt follows a second order DL model:
yt =α 0 + δ 0 zt + δ1 zt −1 + δ 2 zt − 2 + ut
Let z* denote the equilibrium value of zt and let y* be the
equilibrium value of yt, such that
y =α 0 + δ 0 z + δ1 z + δ 2 z
*
*
*
*
The change in y*, due to a change in z*, equals the long-run
propensity times the change in z*
This gives an alternative way of interpreting the LRP
39
Distributed Lag Models
yt =𝛼𝛼0 +𝛿𝛿 0 zt +𝛿𝛿 1 zt-1 + … +𝛿𝛿 r zt–r + ut

In summary, for a temporary change in z, we call δ0 the
impact propensity – it reflects the immediate change in y.

For a permanent change in z, we can call δ0 + δ1 +…+ δr
the long-run propensity (LRP) – it reflects the long-run
change in y after a permanent change.

Question:
How to construct the 95% confidence interval for the LRP?
(homework question)
40
DISTRIBUTED LAG MODELS

Graphical illustration of lagged effects
For example, the effect is biggest
after a lag of one period. After that,
the effect vanishes (if the initial
shock was transitory).
The long run effect of a permanent
shock is the cumulated effect of all
relevant lagged effects. It does not
vanish (if the initial shock is a permanent one).
41
CLASSICAL ASSUMPTIONS

Assumption TS.1 (Linear in parameters)
The time series involved obey a linear relationship. The stochastic processes yt,
xt1,…, xtk are observed, the error process ut is unobserved. The definition of the
explanatory variables is general, e.g. they may be lags or functions of other
explanatory variables.

Assumption TS.2 (No perfect collinearity)
In the sample (and therefore in the underlying time series
process), no independent variable is constant nor a perfect
linear combination of the others.
42
CLASSICAL ASSUMPTIONS

Notation
This matrix collects all the
information on the complete time
paths of all the explanatory
variables
The values of all the explanatory
variables in period number t

Assumption TS.3 (Zero conditional mean)
The mean value of the unobserved
factors is uncorrelated with the values
of the explanatory variables in all
periods
43
CLASSICAL ASSUMPTIONS

Discussion of assumption TS.3
Exogeneity:
Strict exogeneity:

The mean of the error term is uncorrelated to the
explanatory variables of the same period
The mean of the error term is uncorrelated to
the values of the explanatory variables of all
periods
Strict exogeneity is stronger than contemporaneous
exogeneity
TS.3 rules out feedback from the dep. variable on future
values of the explanatory variables; this is often
questionable esp. if explanatory variables “adjust” to past
changes in the dependent variable
 If the error term is related to past values of the explanatory
44
variables, one should include these values as regressors

CLASSICAL ASSUMPTIONS

Unbiasedness of OLS

Assumption TS.4 (Homoskedasticity)
The volatility of the errors
must not be related to the
explanatory variables in any of
the periods
A sufficient condition is that the volatility of the error
is independent of the explanatory variables and that
it is constant over time
 In the time series context, homoskedasticity may also
be easily violated, e.g. if the volatility of the dep.
variable depends on regime changes

45
CLASSICAL ASSUMPTIONS

Assumption TS.5 (No serial correlation)
Conditional on the explanatory variables, the
unobserved factors must not be correlated over
time

Discussion of assumption TS.5
Why was such an assumption not made in the crosssectional case?
 The assumption may easily be violated if, conditional
on knowing the values of the indep. variables, omitted
factors are correlated over time
 The assumption may also serve as substitute for the
random sampling assumption if sampling a crosssection is not done completely randomly
 In this case, given the values of the explanatory
variables, errors have to be uncorrelated across crosssectional units (e.g. states)

46
CLASSICAL ASSUMPTIONS

OLS sampling variances
Under assumptions TS.1 – TS.5:

The same formula as in
the cross-sectional case
Unbiased estimation of the error variance
47
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Gauss-Markov Theorem
Under assumptions TS.1 – TS.5, the OLS estimators
have the minimal variance of all linear unbiased
estimators of the regression coefficients
 This holds conditional as well as unconditional on the
regressors


Assumption TS.6 (Normality)
This assumption implies TS.3 –
TS.5
independently of

Normal sampling distributions

Under assumptions TS.1 – TS.6, the OLS estimators
have the usual normal distribution (conditional on
). The usual F- and t-tests are valid.
48
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA


Example: Static Phillips curve
Discussion of CLM assumptions
TS.1:
TS.2:
Contrary to theory, the estimated
Phillips Curve does not suggest a
tradeoff between inflation and
unemployment
The error term contains factors such
as monetary shocks, income/demand
shocks, oil price shocks, supply
shocks, or exchange rate shocks
A linear relationship might be restrictive, but it should be a good
approximation. Perfect collinearity is not a problem as long as
unemployment varies over time.
49
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Discussion of CLM assumptions
Easily
violated
TS.3:
e.g., past unemployment shocks may lead to future demand
shocks which may dampen inflation; an oil price shock means
more inflation and may lead to future change in unemployment
Assumption is violated if
monetary policy is more
“nervous” in times of high
unemployment
TS.4:
TS.5:
TS.6:
Questionable
Assumption is violated if exchange rate influences persist
over time (they cannot be
explained by unemployment)
50
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Example: Effects of inflation and deficits on
interest rates
Interest rate on 3-months T-bill

Government deficit as percentage of GDP
Discussion of CLM assumptions
The error term represents
other factors that determine
interest rates in general, e.g.
business cycle effects
TS.1: A linear relationship might be restrictive, but it should be a
good approximation. Perfect collinearity will seldomly be a
TS.2: problem in practice.
51
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Discussion of CLM assumptions (cont.)
TS.3:
Easily
violated
e.g., past deficit spending may boost economic activity, which in turn
may lead to general interest rate rises; unobserved demand shocks may
increase interest rates and lead to a change in inflation in future periods
Assumption is violated if higher deficits
lead to more uncertainty about state
finances and possibly more abrupt rate
changes
TS.4:
TS.5:
TS.6:
Questionable
Assumption is violated if business
cylce effects persist across years (and
they cannot be completely accounted
for by inflation and the evolution of
deficits)
52
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Using dummy explanatory variables in time
series
Children born per
1,000 women in year t

Tax
exemption in
year t
Dummy for World
War II years (1941-45)
Dummy for availabity of contraceptive pill (1963-present)
Interpretation
During World War II, the fertility rate was
temporarily lower
 It has been permanently lower since the introduction
of the pill in 1963

53
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Time series with trends
Example for a time
series with a linear
upward trend
54
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Modelling a linear time trend
Abstracting from random deviations, the dependent
variable increases by a constant amount per time unit
Alternatively, the expected value of the dependent
variable is a linear function of time

Modelling an exponential time trend
Abstracting from random deviations, the dependent
55
vari-able increases by a constant percentage per time
unit
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Example for a time series with an exponential
trend
Abstracting from
random deviations,
the time series has
a constant growth
rate
56
BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Using trending variables in regression analysis
If trending variables are regressed on each other, a
spurious relationship may arise.
 Often both will be trending because of other
unobserved factors.
 In this case, it is important to include a trend in the
regression


Example: Housing investment and prices
Per capita housing investment
Housing price index
It looks as if investment and
prices are positively related
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BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA

Example: Housing investment and prices (cont.)
There is no significant relationship
between price and investment
anymore
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DETRENDED SERIES

Basically, the trend has been partialled out
(Recall the partitioned regression)
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Detrending

An advantage to actually detrending the data (vs. adding a
trend) involves the calculation of goodness of fit.

Time-series regressions tend to have very high R2

𝑆𝑆𝑆𝑆𝑆𝑆/(𝑛𝑛 − 1) an unbiased estimator of Var(𝑦𝑦𝑡𝑡 )?

The R2 from a regression on detrended data better reflects how
well the xt’s explain yt.
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Seasonality and Deseasonalizing the Data





Often time-series data exhibits some periodicity, referred
to seasonality.
Example: Quarterly data on retail sales will tend to
jump up in the 4th quarter.
Seasonality can be dealt with by adding a set of seasonal
dummies.
As with trends, the series can be seasonally adjusted
before running the regression.
Deseasonalizing the variable can be obtained from the
residuals from the regression of the variable on the
seasonal dummies.
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BASIC REGRESSION ANALYSIS WITH
TIME SERIES DATA
Modelling seasonality in time series
 A simple method is to include a set of seasonal
dummies:

= 1 if obs. from december
= 0 otherwise
The regression coefficients on the explanatory variables
can be seen as the result of first deseasonalizing the dep.
and the explanat. variables
 R2 based on first deseasonalizing the dep. var. may better
reflect the explanatory power of the explanatory variables

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