19.06.2023 Tutorial 4#
Hodrick-Prescott Filter#
a) disaggregate Trend and Fluctuations in GDP
\[
\min \{ \bar{y}\}^T_{t=1}
\bigg(
\underbrace{\sum_{t=1}^T (y_t-\bar{y_t}^2)}
_{\text{change in trend}}
+ \lambda \cdot \underbrace{\sum_{t=2}^{T-1}
[(\bar{y}_{t+1} - \bar{y_t}) - (\bar{y}_t - \bar{y}_{t+1})]}
_{\text{cyclical component}}
\bigg)
\]
If \(\lambda = 0\): Result is equal to observations (pure cycle)
if \(\lambda = \inf\): Result is straight line
b) find differenct values
\[\begin{split}
\begin{aligned}
Y_t &= \; \bar{Y}_t &\cdot Y_t^c \\
\log Y_t &= \log \bar{Y}_t &\cdot \log Y_t^c \\
y_t &\equiv \; \bar{y} &\equiv c_t
\end{aligned}
\end{split}\]
Covariance#
a) the GDP is most volatile
b) Differences in deviations
common sample deviation: \(\sqrt{\frac{ 1 }{N} \sum ...}\)
sample standard deviation: \(\sqrt{ \frac{ 1 }{N \bf{-1}} \sum}\)
c) Covariance
when positive correlation: Cov is positive
negative vice-versa d) Correlation
\[
\rho = \frac{ Cov (x,y) }{\sqrt{var(x)* var(y)}} \in [-1,1]
\]
while regression: \(y = a+bx+e\), with
\[
\hat{b} = \frac{ cov(x,y) }{var(x)}
\]
e) Calculation of sample correlation:
\[\begin{split}
Cov (GDP, Cons.) = \frac{1 }{4-1}
\big( (\overbrace{-0.01}^{x_1}-\overbrace{0}^{x_{mean}} )
* (\overbrace{-0.06}^{y_1}-\overbrace{0}^{{y_{mean}}}) + ... \big)
= pos. value\\
Corr = \frac{ Cov}{\sqrt{var(x)*var (y)}} = 0.999
\end{split}\]