# Estimating Beta from unevenly spaced price history

I have a certain non-stock asset that has 1 transaction every 1 to 8 months.
I also have a price index of that class of asset compiled by another party on monthly basis.
If I regress $price = \alpha' + \beta' index$, the $R^2$ is 0.975 to 0.999 for every asset.

How do I obtain an estimate of $\beta$ of CAPM $R_i = R_f + \beta(R_m-R_f)$

Right now, I am blindly guessing it through the following steps:
1. Regress $price = \alpha' + \beta' index$, obtain $\alpha'$ and $\beta'$
2. Generate interpolated monthly price $\widehat{price}_t$ by plugging in the monthly index
3. Generate monthly asset return $\hat{R}_{t+1}=\frac{\widehat{price}_{t+1}}{\widehat{price}_t}-1$
4. Generate monthly market return $\hat{R}_{m_{t+1}}=\frac{index_{t+1}}{index_t}-1$
5. Regress $\hat{R_{t}}$ on $\hat{R}_{m_{t}}$ and obtain $\beta$ of CAPM

Is this method valid? If not, what would be the proper convention?

• There's really no proper convention here. There are a lot of different options that might be better in some cases than others. Also, how much effort you put in might depend on what you're trying to do and what your boss wants. – John Apr 21 '14 at 17:11

## 1 Answer

No, it's not. First, what you ought to be regressing are returns, not prices. Second, by interpolating you're underestimating the variance of the asset price in the interval between index price observations. Through your choice of interpolation method, you're essentially picking an arbitrary price in the middle.

What you ought to be doing is maximum likelihood estimation (MLE). You'll have to assume a parameterized family of joint stochastic processes and estimate the parameters given the price observations. Whenever you don't have synchronous data, you'll have a probability distribution for the missing price conditional on all other data points (in its future and in its past). Hence the distribution you'll be using to maximise the likelihood of the observed price will be wider than otherwise.

• This sounds like the same problem faced when doing model fitting on tick and order book data - do you have any handy references to the conversion from simple regression to using proper MLE when transitioning to asynchronous event data? – experquisite Apr 22 '14 at 13:49