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\begin{document}

\title{\bf The Slutsky Effect}
\author{\Large Carlo H\"{a}m\"{a}l\"{a}inen \\ {\texttt carlo.hamalainen@gmail.com} }
%\date{}
\maketitle

The {\em Slutsky effect}, as understood by many economists, is
described in the following quote from Barnett (2006):

\begin{quote} If the variables that were taken to represent business
cycles were moving averages of past determining quantities that
were not serially correlated – either real-world moving averages
or artificially generated moving averages – then the variables of
interest would become serially correlated, and this process would
produce a periodicity approaching that of sine waves.  \end{quote}

To demonstrate this effect we take a sequence of independent identically
distributed random variables with mean $\mu$ and variance $\sigma^2$,
compute a moving average, and look for oscillatory behaviour. (The
example that follows is similar to Royama p.~131.)

Suppose that $u_i$ is a sequence of independent identically distributed
random variables . Here we generate $1000$ samples, shown in
Figure~\ref{figUseries}.

\begin{sageblock}
set_random_seed(0)

def avg(L):
    return sum(L)/(1.0*len(L))
    
def variance(L):
    mu = avg(L)
    return avg([(x - mu)**2 for x in L])    

nr_samples = 1000
N = 10 # u_i is in [0, 1, ..., N - 1]

U = [randrange(N) for _ in range(nr_samples)]

U_mu = avg(range(N))
U_var = variance(range(N))
\end{sageblock}

\begin{figure}[htb]
\begin{center}
\sageplot[width=.75\textwidth]{list_plot(U[:100], ymin = 0, plotjoined = True, marker = 'o', axes_labels = ["time", "u_i"], figsize = 5, rgbcolor = 'black')}
\end{center}
\caption{First 100 points from sequence $u_i$.}
\label{figUseries}
\end{figure}

There is no trend nor systematic behaviour of the series. We can check this by calculating the autocorrelation $R(t)$:
\[
R(t) = \frac{\mathbb{E}((u_i - \mu)(u_{i+t} - \mu))}{\sigma^2}.
\]
The autocorrelation ranges between $-1$ and $1$, meaning perfectly
anti-correlated and perfectly correlated, respectively.

\begin{sageblock}
def autocorrelation(L, t):
    """
    Biased estimator of the autocorrelation.
    """
    
    mu = avg(L)
    
    return avg([(L[i] - mu)*(L[i+t] - mu)
        for i in range(len(L) - t)])/variance(L)
    
t_max = 20

U_ac = [autocorrelation(U, t) for t in range(t_max)]
\end{sageblock}

\begin{figure}[htb]
\begin{center}
\sageplot[width=.75\textwidth]{list_plot(U_ac, plotjoined = True, marker = 'o', ymin = 0, rgbcolor = 'black')}
\end{center}
\caption{Autocorrelation of the sequence $u_i$.}
\label{Uautocorr}
\end{figure}


This is as expected. In Figure~\ref{Uautocorr}
We see that $R(0) = 1$ because any sample is autocorrelated with itself, and $R(t)$ is around $0$ for all $t > 0$ since the samples are independent. So what happens if we take a moving average of the series? Define a new random number $w_t$ by
\[
w_t = \frac{1}{k}\sum_{i = 1}^{k} u_{t - i + 1}.
\]
Below we compute this new series for $k = 10$ and plot the autocorrelation.

\begin{sageblock}
moving_avg_len = 10

W = [(1.0/moving_avg_len)*sum(U[i:i+moving_avg_len])
    for i in range(len(U) - moving_avg_len)]

p = list_plot(W, plotjoined = True, marker = 'o', figsize = 5, 
    rgbcolor = 'black')
p.save(filename = "W_plot.pdf", ymin = 0)
\end{sageblock}

\begin{figure}[htb]
\begin{center}
\includegraphics{W_plot}
\end{center}
\caption{Sequence $w_i$.}
\label{firstAvg}
\end{figure}

Figure~\ref{firstAvg} shows oscillatory motion. This is because two points
that are $j$ points apart share $k-j$ points of the original series
(if $j < k$) and $0$ otherwise. We see this in the autocorrelation,
calculated as follows, and shown in Figure~\ref{firstauto}.

\begin{sageblock}
W_ac = [autocorrelation(W, t) for t in range(t_max)]
\end{sageblock}

\begin{figure}[htb]
\begin{center}
\sageplot[width=.75\textwidth]{list_plot(W_ac, plotjoined = True, marker = 'o', figsize = 5, ymin = 0, rgbcolor = 'black')}
\end{center}
\caption{Autocorrelation of the sequence $w_i$.}
\label{firstauto}
\end{figure}

If we repeat the taking of a moving average then we see (Figure~\ref{fouravg})
what appears to be very regular behaviour:

\begin{sageblock}
W1 = [(1.0/moving_avg_len)*sum(W[i:i+moving_avg_len])
    for i in range(len(W) - moving_avg_len)]

W2 = [(1.0/moving_avg_len)*sum(W1[i:i+moving_avg_len])
    for i in range(len(W1) - moving_avg_len)]

W3 = [(1.0/moving_avg_len)*sum(W2[i:i+moving_avg_len])
    for i in range(len(W2) - moving_avg_len)]

W3_plot = list_plot(W3, plotjoined = True, marker = 'o',
    figsize = 5, rgbcolor = 'black')
W3_plot.save("W3_plot.pdf", ymin = 0)
\end{sageblock}

\begin{figure}[htb]
\begin{center}
\includegraphics{W3_plot}
\end{center}
\caption{Sequence from four moving averages.}
\label{fouravg}
\end{figure}

It is interesting (but not surprising?) that the points in
Figure~\ref{fouravg} stay near the mean of the original,
$\sage{avg(U)}$.

\clearpage

References:
    
\begin{itemize}    
\item Barnett, 2006. "Chancing an interpretation: Slutsky's random
cycles revisited ," European Journal of the History of Economic
Thought, Taylor and Francis Journals, vol. 13(3), pages 411-432,
September.

\item Royama, 1992. Analytical population dynamics. Springer.

\item Slutsky, 1937. The summation of random causes as the source of cyclic processes. Econometrica 1937;5:105-46

\end{itemize}

\end{document}