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I am working on time series prediction. I have two data sets $D1=\{x_1, x_2,....x_n\}$ and $D2=\{x_n+1, x_n+2, x_n+3,...., x_n+k\}$. I have three prediction models: $M1, M2, M3$. All of those model are trained using samples in data set $D1$, and their performance is measured using the samples in data set $D2$. Let say the performance metrics is MSE (or anything else). The MSE of those models when measured for data set $D2$ are $MSE_1, MSE_2, $ and $MSE_3$. How can I test that improvement of one model over another is statistically significant.

For example, let say $MSE_1=200$, $MSE_2=205$, $MSE_3=210$, and total number of sample in data set $D2$ based upon which those MSE are calculated is 2000. How can I test that $MSE_1$, $MSE_2$, and $MSE_3$ are significantly different. I would greatly appreciate if anyone can help me in this problem.

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One of the linked posts above alludes to using a likelihood ratio test, although your models have to be nested in one another for this to work (i.e. all the parameters in one of the models must be present in the model you are testing it against).

RMSE is clearly a measure of how well the model fits the data. However, so is likelihood ratio. The likelihood for a given person, say Mrs. Chen, is the probability that a person with all her parameters had the outcome she had. The joint likelihood of the dataset is Mrs. Chen's likelihood * Mrs. Gundersen's likelihood * Mrs. Johnson's likelihood * ... etc.

Adding a covariate, or any number of covariates, can't really make the likelihood ratio worse, I don't think. But it can improve the likelihood ratio by a non-significant amount. Models that fit better will have a higher likelihood. You can formally test whether model A fits model B better. You should have some sort of LR test function available in whatever software you use, but basically, the LR test statistic is -2 * the difference of the logs of the likelihoods, and it's distributed chi-square with df = the difference in the number of parameters.

Also, comparing the AIC or BIC of the two models and finding the lowest one is also acceptable. AIC and BIC are basically the log likelihoods penalized for number of parameters.

I'm not sure about using a t-test for the RMSEs, and I would actually lean against it unless you can find some theoretical work that's been done in the area. Basically, do you know how the values of RMSE are asymptotically distributed? I'm not sure. Some further discussion here:

http://www.stata.com/statalist/archive/2012-11/index.html#01017

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This answer doesn't take the fact into account, that your data form a time series but I don't think this would be a problem.

When using RMSE, this post suggests using a t-test: Testing significance of RMSE of models

You could also use Pearson's correlation to assess your fit. According to this post, you can use Wolfe's t-Test for that: Statistical significance of increase in correlation

I currently trying to learn about the same problem. I would appreciate more detailed answers myself.

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There are two main ways to do this, but first I’ll challenge the idea that you want to pick only one. Most likely, an ensemble model of the three separate models will achieve the best performance of all.

The main, perhaps best, way to do it is to use the model to obtain confidence intervals around the evaluation metric. This is commonly done via bootstrapping (or Poisson bootstrap).

The other way is to use a statistical test. Every test makes different assumptions, and these are often used to compare a value or sample taken from a distribution rather than a single point evaluation. Many of these statistical tests formally require independence, which you usually don’t have when comparing multiple results of the same model or multiple models over time series data.

With time series prediction specifically, you should be doing backtesting with cross-validation and evaluating train and test error at each time (example). When you do this, I doubt your models will all perform so similarly that you need a statistical test to differentiate; most likely, you’ll see large differences.

Note also that historical evaluation metrics (comparing actuals to forecast) alone are insufficient for prediction evaluation. Given two predictions that fit known historical data perfectly but one also matches prior beliefs about the future and the other clearly violates (e.g., if one vanishes to zero but you have reason to believe that can’t happen), you’ll prefer the prediction that better matches your prior.

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