Contact time series analysis is only half a year, try your best to answer. If the answer is wrong, please point it out.

For the first question, we divide it into the following two questions:

Why is it fixed? (Why smooth? )

Why is it weakly stationary? (Why is it weak and stable? )

Why is it fixed? (Why smooth? )

For every statistical problem, we need to make some basic assumptions. For example, in the univariate linear regression (), we should assume that ① is irrelevant and non-random (it is a fixed value or is considered to be known) and ② the independent and identical distribution obeys the normal distribution (the mean value is 0 and the variance is constant).

In time series analysis, we consider many reasonable assumptions, which can simplify the problem. The most important assumption is stationarity.

The basic idea of stationarity is that the probability law of controlling process behavior will not change with time.

The basic idea of stationarity is that the behavior of time series does not change with time.

Therefore, we define two kinds of stationarity:

Strict stationarity: For all choices of natural number n, all choices of time points, and all choices of delay K, the joint distribution of, and is the same as that of, and, then a time series {} is said to be strictly stationary.

Strongly stationary process: For all possible N, all possible K, when the joint distribution of,, and is the same as the joint distribution of,, and, we call it strongly stationary.

Weak stationarity: A time series is said to be weakly (second order or covariance) stationary if the following conditions are met:

(1) the mean function remains constant with time, and

② γ(t，t？ K) = γ(0, k) for all time t and lag k.

Weakly stationary process: When ① the mean function is a constant function and ② the covariance function is only related to the time difference, we call it weakly stationary.

At this point, we turn to the second question: Why is it weakly stationary? (Why is it weak and stable? )

Let's talk about two smooth differences:

There is no relationship between the two stationary processes, that is, weak stationarity is not necessarily strong stationarity, and strong stationarity is not necessarily weak stationarity.

On the one hand, although it seems that the requirement of strong stationarity is stronger than that of weak stationarity, strong stationarity is not necessarily weak stationarity, because its moment does not necessarily exist.

For example, {} independently obeys Cauchy distribution. {} is strongly stationary, but not weakly stationary, because the expectation and variance of Cauchy distribution do not exist. It doesn't exist because it is not absolutely integrable. )

On the other hand, weak stationarity is not necessarily strong stationarity, because the second moment property cannot determine the distribution property.

Example: independence from each other. This is weakly stationary, but not strongly stationary.

Knowing the root of these differences, we can also write some connections between them:

When the first moment and the second moment exist, the strongly stationary process is weakly stationary. (The condition can be simplified to the existence of second moment, because)

When the joint distribution obeys multivariate normal distribution, the two stationary processes are equivalent. (The second moment of multivariate normal distribution can determine the distribution properties)

The reason why weak stationarity is used instead of strong stationarity is mainly because the condition of strong stationarity is too strong, both in theory and in practice.

Theoretically, it is generally difficult to prove that a time series is strongly stationary. As the definition says, we should compare all possible joint distributions of n, all possible k, when,, and, are the same. When the distribution is complex, it is not only difficult to compare all the possibilities, but also difficult to write its joint distribution function.

In fact, for data, we can only estimate their mean and second moment, and we can't know their distribution. So we use ACF in future model construction and prediction, and these attributes are related to weaknesses and attributes. Moreover, the professor who taught me time series said, "The general linear process (weak, linear and causality) covers about 10% of the real data." If we consider strong stationarity, I think it may not even be 5%.

For the second question:

One day when the professor was reviewing his undergraduate thesis, he saw a financial writer using stationary time series to estimate the stock trend (I really don't know what this guy thinks). At that time, the professor said: "Many things in the financial field are difficult to estimate because they often mutate and are not stable at all."

Sure enough, in the final practice stage of the paper, the correct rate of stock selection is 40%. Even lower than the expected 50% (after any point, it is either up or down).

During the summer vacation, I used some time series methods to try to develop a programmed trading program.

At first, the rate of return was not bad, but later, more and more ... the direct loss behind it ... (the software is the pyramid, and the second column is the profit rate)

The loss chart was not cut at that time, and it can't be made up now. This program has been deleted.

So it should have nothing to do with stability, after all, my approach is not to assume stability. If it goes well, I won't be unprofitable in the future.

(Tucao) I'm really not suitable for stocks and futures ... too high-end to understand. ...

exceed