Help w/ finding correlation between unemployment rate and home prices over the same time period? I am interested in analyzing the correlation between nationwide home prices and nationwide unemployment rates, both of which are leading economic indicators. I have data on nationwide home prices by using the Case-Shiller nationwide home price index (found here: http://us.spindices.com/indices/real-estate/sp-case-shiller-us-national-home-price-index), and I have data on nationwide unemployment rates from the Bureau of Labor Statistics. 
Preliminary hypothesis/background info: Home prices are high when economy is doing well, and unemployment rates are low when the economy is doing well. So common sense tells me that as the unemployment rate rises, then the Case Shiller home price index decreases, which means there should be a negative correlation. But I don't know how to prove this. Here is a summary of the data I have:
I have the data for the Case-Shiller nationwide Home Price index for every month over the last ten years (1/1/2005-12/31/2014) which means 120 data points. I also have all the data for the nationwide Unemployment Rate over the same time period (1/1/2005-12/31/2014), which also means 120 data points. Both data are collected for the end of the month over the same time period, which means there is zero lag in the data sets.
What kind of correlation analysis do I need to do to determine if there is any correlation between these two data sets? Cross-correlation? Time-series analysis?
Thank you so much for any advice on how to start this research! Any help on what direction I should go would be incredibly appreciate. 
Thank you!
 A: Easiest way to show is to plot housing price index against unemployment rate and that should be a downward slopping curve. If you need a single number then you should calculate the correlation coefficient.

x bar: Mean of Unemployment Rate
y bar: Mean of Housing price index
s : Standard Deviation
Obviously, unemployment and housing prices are correlated. But that does not mean unemployment causes housing prices to rise or vice versa. Both unemployment and housing prices are driven same underlying macroeconomic variable.

The assumption in economic literature is that the -effects in housing markets precede the effects in labor market. So it is not uncommon to see unemployment as a function of housing price index
But you would need more variables to tease out the effect of housing prices on unemployment. The following study concludes that --A 10% appreciation in house prices yields to a 3.4% decrease in the unemployment rate.
http://www.cepii.fr/PDF_PUB/wp/2014/wp2014-25.pdf
So your best bet is to find more variables to control for confounding effects; then use a regression approach
Example: 
Unemployment Rate = F(housing prices, other control variables)
There are definitely auto correlation issues as Richard mentioned, and if you want to do an in-depth analysis of the relationship between unemployment and housing prices, you have to be rigorous. 
@ Stas K: I agree with your point. But I would be hesitant to establish causality here. I would leave it at the correlation level unless I have more variables to control for confounding effects.
A: Correlation is insufficient for this analysis. You need to perform an appropriate time series vector autoregression analysis regressing variables on each other and lagged values (from previous months). Then you can test these old lags to establish Granger causality.
A: Even though correlation can be calculated (it is technically possible), that does not automatically imply it makes sense. @StasK proposed a more suitable solution than simple correlation analysis, but he did not explain the motivation. I will try to do that briefly.
The problem with prices and unemployment is that they are not independently and identically distributed (not i.i.d.) The unemployment this month is closely related to the unemployment last month (the month-to-month change is usually quite small, except for seasonal effects). Meanwhile, it is much less related to unemployment 100 months ago. The same argument also applies to prices. 
Correlation of non-i.i.d. variables does not measure what it measures for i.i.d. data (where it is a suitable metric); it cannot be interpreted the regular way. Neglecting the non-i.i.d. nature of the data is likely to lead to false (or at least unreliable) conclusions. Building a time series model is a better alternative as time series models are intended for non-i.i.d. time-dependent data. @StasK proposed a particular model that could be used. 
If you have not heard much of VAR models before, you may need some time to get going. If you do not have the time, at least be aware that simple correlation is not really applicable to the data you have.
