12 votes
Accepted

Relationships between white noise and random walk

I will assume a white noise is a process $(\varepsilon_t)$ with zero mean, no autocorrelation and constant variance $\sigma^2 > 0$ while a random walk is a process $(x_t)$ defined by $$ x_{t+1} =...
  • 3,866
7 votes

Estimate covariance matrix using prices

If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (...
  • 8,139
6 votes

Are cumulative returns stationary?

Hi: Even if returns were stationary ( which is probably dependent on the time series one is considering ), cumulative returns, where $n$ is not fixed ( as it in say a rolling sum with a fixed window ...
  • 1,032
5 votes
Accepted

How is stock data objectively different to this random walk?

I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is ...
  • 27.1k
5 votes

Is a stationary process necessarily mean-reverting?

The concept of 'mean reversion' is tricky in continuous time. Most people would call 'mean reverting' a process where the drift pulls back towards a long run mean, and I assume that this is what you ...
  • 4,247
5 votes

Relationships between white noise and random walk

Regarding the relationship between white noise and a random walk, I would put it this way: a random walk is integrated white noise. [And vice versa we get a white noise when we differentiate/...
  • 9,202
5 votes

log return of sp500. Stationary vs strictly stationary

We can talk about whether a strictly stationary or weakly stationary process might usefully describe that data. My answer to both would be yes. I also have issues with other text that people have ...
  • 6,484
5 votes

Differencing vs Detrending financial time series

Hi: It depends on what the DGP of the original process is. Is the process trend stationary or difference stationary ? If it's trend stationary then de-trending is the way to go. If it's difference ...
  • 1,032
4 votes
Accepted

Does predictability in a VAR process imply mean reversion or momentum?

The point of confusion may be in thinking that a predictable price process is synonymous with a mean-reverting process while using the definitions in these papers, it's actually the opposite! In the ...
  • 6,484
4 votes

Are cumulative returns stationary?

Stock prices are definitely not stationary as tomorrows closing price is strongly influenced by today's closing price and prices tend to change. Returns can be potentially stationary and are therefore ...
  • 7,739
3 votes
Accepted

Does Weak stationarity imply ergodicity ?

Ergodicity is connected to mixing, meaning there is one limiting distribution and it is used for time averages too. If you take a process in the real numbers that starts at a random value and then ...
3 votes

Is a stationary process necessarily mean-reverting?

Let's consider the following example: the process is initialized randomly with $\pm1$ and then stays there forever. Seems stationary to me, but it would never cross its mean.
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3 votes

How to use autocorrelation plot to interpret time series data?

Just by looking at the graphs, I'd say: Unit root Constant series Seasonality AR model No AC No AC
  • 627
3 votes

Issues making series stationary

Your shift is in the wrong direction. Do this: df.price = pd.to_numeric(df.price) df['logret'] = np.log(df.price/df.price.shift(1))
3 votes
Accepted

Transforming a time series

Fractional differentiation (or differencing) is a technique that transforms an input series to a stationary series while retaining "long-term" memory. Consider the following example based on ...
2 votes
Accepted

Does unit root stationary imply mean stationary and variance stationary?

Consider the following AR(1) process with i.i.d. normal errors that have zero mean and finite variance $\sigma^2>0$, $$ x_t = (1-\rho)\mu + \rho x_{t-1} + \epsilon _t$$ Now suppose $ \rho = 1/2$ ...
  • 527
2 votes

Why is OU process stationary?

I think you misunderstood the definition. Be stationary does not mean not depend of the time as you can check here. (Sorry for putting an wikipedia link here as I suppose you may have read it) ...
  • 608
2 votes

How is stock data objectively different to this random walk?

For both time-series, just plot the log returns. You will see that one is not a Random-Walk .. the S&P500 since you will get values that far beyond the normal distribution. Just watch this video ...
  • 493
2 votes

Problem - stationarity and relevance

Any data transformation to assure stationarity eliminates part of the signal in many cases the signal is not completely eliminated so you can still perform the required analyses but in some others as ...
  • 436
2 votes

Stationary Process with autocorrelation in Variance; square root rule

You are correct in that the series is not stationary. The ADF test isn't designed to test for stationarity outside the center of location. You are not going to be able to use the square root rule to ...
  • 4,119
2 votes
Accepted

Would you consider yield a stationary or non-stationary process?

Following Meucci (Risk and Asset Allocation book, page 112-113) you should use "change of yield to maturity" (simple change, not percentage) since they represent Fixed Income´s invariant. Change of ...
  • 296
2 votes

Does forecasting asset returns by default assumes non-stationarity of asset returns?

This looks confused? I don't understand what you're saying in the second paragraph... Comment 1: "Best" forecast depends on what you mean by "best." Let $Y$ be a random variable and $\mathcal{F}$ be ...
  • 6,484
2 votes

Differencing vs Detrending financial time series

Let me try to write formulae to explain the differences: When $X_t=a+b\,t + c\,\xi_t$, where $\xi_t$ is an iid centered and reduced noise (ie $\mathbb{E}\xi=0$ and $\mathbb{E}\xi^2=1$. With $X_(t+1)-...
  • 10.9k
2 votes
Accepted

Estimate covariance matrix using prices

For the same reason you can't meaningfully measure covariance/correlation using price of individual assets...correlation (covariance by extension) represents the comovement in deviations from ...
  • 1,598
2 votes
Accepted

Do stationary prices need to be differenced for VaR?

I worry that power prices are very unlikely to be stationary. It is possible the mean does not vary wildly over time, and the price process may not be integrated, i.e. prices may not require ...
  • 2,810
2 votes
Accepted

Are cumulative returns stationary?

In a nutshell... It's always prudent and conservative to assume that prices are non-stationary. But it's not actually as obvious this is true as it intuitively sounds. Intuitively, any random walk ...
  • 4,921
1 vote
Accepted

How can I approximate Dollar Bars from Minute Data instead of Tick Data?

The following python package, mlfinlab, provides an implementation for both standard and information-driven bars. The good news is that you won't have to implement the techniques from scratch and they ...
1 vote

log return of sp500. Stationary vs strictly stationary

They probably can be modelled using a weakly stationary process. To quote Section 1.2.1 from these lecture notes: [Asset] returns [...] typically fluctuate around a constant level, suggesting a ...
  • 200
1 vote

Does forecasting asset returns by default assumes non-stationarity of asset returns?

If we assume the assets returns are stationary then the best forecast can only be the mean of the distribution. This part is not accurate. Stationarity, even in its strongest sense, only implies that ...
1 vote
Accepted

principal component analysis on non stationary data

You may know that they are two definitions of stationary (see for instance Series of Irregular Observations: Forecasting and Model Building; this book probably contains all you need to model ...
  • 10.9k

Only top scored, non community-wiki answers of a minimum length are eligible