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In my understanding, a time series model is a way to predict the observation. For example, having a time series data points, one possible way to model is using AR model which is $y_{t}$ by $y_{t}= a_{1}x_{t}+a_{2}x_{t-2}+a_{3}x_{t-3}+...+\epsilon_{t} $.

There are many applications in real life using time series series model. For example, it seems when people are talking about prediction in finance, it is usually refers to time seres model.

However, I wonder why would the time series model would work? It seems really strong assumption on betting the future observation would be a combination of the disturbance, weighted by the coefficients $a_i$.

I know that there would be assumptions like stationarity and homoscedasticity (although heteroscedasticity would still keep it unbiased) to ensure the unbiased estimation for the coefficients. However, I guess my problem is at bigger level of asking under what situations the time series model can capture the essence of the target it tries to model.

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    $\begingroup$ Just a little terminology quibble but what you describe looks like an autoregressive (AR) model to me. It's one form of time series modeling. A time series is simply a series of values measured over time, not a model or something that “works”. $\endgroup$ – Gala Jul 24 '13 at 19:35
  • $\begingroup$ Thanks Gael for your comment. Yes I would modify my post. Thanks for your reminding. $\endgroup$ – Jack2019 Jul 25 '13 at 16:47
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You use a time series model because it makes fewer assumptions about the data than the alternative. Most other types of statistical models assume independent observations: each case can be treated separately to every other case, and they don't affect one another. This is a very strong assumption, and one that is commonly violated for data that has a time component. If there is positive autocorrelation, then if a data point is high, then subsequent data points are also likely to be high; conversely, if there is negative autocorrelation, then a data point that's high is likely to have subsequent data points that are low. Similarly, you may have seasonality effects: January's figures may be consistently higher than July's figures, all other things being equal.

Time series models allow you to take this behaviour into account when making predictions. Different types of models will do it in different ways. Some models can also let you include independent variables, much like a normal regression analysis; these independent variables can be fixed or time-varying.

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A time series model is usually used when the data used are ordered after then the observation occured. The most common time series models regress on their own previous values (autoregressive (AR) or moving average (MA)) or previous value of itself and previous values of other time series variables (vector autoregression (VAR)). Time series model work in the same way any other regression work, when there is a relationship between the dependent and the independent variables. But there are some point to what you are saying, I think you will find it interesting to look up 'the Lucas Critique'.

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I like Hong Ooi's answer. All I want to add is that time series forecasting often works in practice because things tend not to change much in the short term. For example, I don't know whether it will rain today. But it's sunny now, and it is very likely to be sunny 15 minutes from now. Or if you want to forecast the number of tourists visiting the country for next month, it will probably be roughly the same as last August. The number of tourists is increasing over time, and a forecast might take that trend into account, but even if it doesn't, it will still be pretty accurate.

We have to make these kinds of predictions all the time in our everyday lives. The only data we have, by definition, is what happened in the past. The best (and only) forecasts we have are the ones which predict the future based on the past. Time series methods are just a formal way of doing this. And ultimately, even when you run the most sophisticated models, the answer to the question "what will happen tomorrow?" often seems to be "roughly the same as what happened yesterday."

Your question is: what things cannot be predicted from time series? Many people would say the stock market. It's true that I can't predict tomorrow's closing prices. But in the very short term, the stock market is predictable, as the trader Jesse Livermore discovered at the beginning of his career (described in the marvellous book Reminiscenses of a Stock Operator.) Essentially, he was allowed to bet on minute-to-minute stock prices displayed on a ticker tape, and found that he could predict them accurately enough to make a steady income. But when he moved to Wall Street, he found that he could not trade real stocks quickly enough to exploit his knowledge, and went bust.

However, some things, even in the short term, are completely random. For example, there is no way that I can use time series to predict what the result of a coin flip will be, even if I've already flipped the coin many times. If you think about it, this is precisely the "gambler's fallacy". We have a very strong feeling that we can predict the near future based on the past, and it fails us when we are confronted with things which don't depend on the past. But the reason why we have this feeling is precisely because, most of the time, it's correct.

What things in real life behave like this? Maybe earthquakes? Supposedly you can predict how frequently they will happen, but not exactly when.

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