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\begin{document}
\title{Econometrics Notes}
\author[1]{Felicia Cowley}%
\affil[1]{George Mason University}%
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\date{\today}
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4/10/2018
Chapter 10
Time Series Properties
temporal ordering of observations is not arbitrary (pattern)
(\(x_t\) is related to~\(x_{\left\{t1\right\}}\))
~ ~ ~ ~ ~ ~ example: GNP, Dow Jones's financial indices
``sample'' is only one realized path of many possibilities
~~~~~~~~~~~~for example, as inflation increases so does unemployment
but significance doesn't hold up, regardless of Keynes
~ ~ ~ ~ ~ ~ another example, what is the relation between the tax rate
and tax revenue (Laffer Curve) How would you estimate this with OLS?
\par\null
Static Models
current value of one variable is modeled as the resutl of the current
value of explanatory variables
~ ~ ~ ~ ex1~ ~~\(inf_t =\beta_0 +\beta_1 unemp_t +u_t\)
~ ~ ~ ~ ex2~ ~~\(mrdrt_t=\beta_0 +\beta_1 convrte_t +\beta_2 unemp_t +\beta_3 youngml_t +u_t\)
~ ~ ~ ~ ~ ~ ~ ~ ~ we expect the conviction rate~ and unemployment rates
to be positive. Similarly for young male fraction. Maybe this model
needs a lag rate as it might take some people time if unemployed to
commit murders.~~
\par\null
Finite distributed lag models
the explanatory variables are permitted to influence the dependent
variable with a time lag
\(gfr_t=\alpha_0 +\delta_0 pe_t+\delta_1 pe_{t1}+\delta_2 pe_{t2}\)
\(y_t=\alpha_0 +\delta_0 z_t+\delta_1 z_{t1}+...+\delta_q z_{tq} +u_t\)
Transitory shock:~\(\frac{\Delta y_t}{\Delta z_{t5}}=\delta_5\)
Permanent shock:~\(\frac{\Delta y_t}{\Delta z_{tq}} +...+\frac{\Delta y_t}{\Delta z_t}\)
\par\null
OLS for finite samples
Assumption 1) linear in parameters
\(y_t=\beta_0+\beta_1 x_{t1}+\beta_2 x_{t2}+....\)
Assumption 2) no perfect multicollinearity
Assumption 3)~\(E(u_tx)=0\) that is, the expectation of the error
term is zero
~~~~~~~~~~~~~~~~~~~~Exogeneity means that~\(E(u_tx_t)=0\) where as
strict exogeniety is~\(E(u_tx)=0\)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\(E(\beta_j \hat)=\beta_j\)~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~\(Var(u_t  x)=var(u_t)=\sigma^2\)
\(Cor(u_t ,u_s)=0\)
requires given the explanatory variables, errors have to be
uncorrelated
\(Var(\beta_j \hat x)=\frac{\sigma^2}{SST_j (1R_j^2)}\)
\par\null
Using dummy variables in time series
\(\hat {gfrt} = 98.68 +0.083 pe_t24.24 WWII_t\)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~\((3.21) (.030)\)~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~\((7.46)\)
All variables are statistically significant as n=72 and r squared is
.473
the best way to format this would be involving a dummy variable so for
instance zeros would be for years leading up to the pill but ones after.
\par\null
Chapter 11
Assumptions of the static and finite models are very restrictive (and
likely unrealistic for most ``real world'' problems)
~ ~ strict exogeneity
~ ~ homoskedasticity over time
~ ~ no serial correlation
~ ~ These are unlikely in a time series
\par\null
Stationary stochastic process
a stochastic process (ie,~\(x_t :t=1,2...\)) is stationary if every
collections of indices~\(1 \leq t_1 \leq t_2 \leq ...t_n\) the joint distribution of
(\(x_{t1}, x_{t2}, ..., x_{tn}\)) is the same as that of (\(x_{t1+h}, x_{t2+h}, ..., x_{tn+h}\)) for
all integers~\(h \geq 1\)
Covariance stochastic process
\par\null
Weakly dependent series
a stochastic process is weakly dependent if~\(x_t\) is
``almost independent'' of~\(x_{t+h}\) if h grows to infinity
(for all t)
two examples of weakly dependent series:
moving average process of order 1 (MA1)
~ ~ ~ ~ \(x_t=e_t+\alpha_1 e_{t1}\)~ iid series e\_t
~ ~ ~ ~ ~ ~ ~ ~ weakly dependent because observations that are more
than one time period apart have nothing in common and are therefore
uncorrelated
autoregressive process order 1 (AR1)
~ ~ ~ ~ ~ ~ \(y_t=\rho_1 y_{t1} +e_t\)~ ~ ~ ~ ~ ~ ~ ~ ~ If
stable~\(e_1<1\) holds
``Random walk'': called random walk because it wanders from the previous
position by a random amount ``\(e_t\)''
~~~~~~~~~\(y_t=y_{t1} +e_t\)
\par\null
Potential Multiple choice:
cross sectional data vs time series data
~ ~ ~ ~ time series is based on temporal ordering
stochastic process refers to sequence of random variables indexed by
time
sample size for time series dataset is the number of time periods over
which we observe the variables of interest~
static model is postulated when the change in the independent variable
time t is believed to have an immediate effect on the dependent
variable; no lag term
if an explanatory variable is strictly exogenous it implies that the
variable cannot react to the dependenton to what has happend int he past
seasonal adjusted series
a process is stationary if any collection of random variables in a
sequence is tkaen and shifted ahead by h time periods but the joint
probabilitiey distrubtion remains unchanged

\par\null
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