Yes, there is some hypothesis testing for that.

If any of your coefficients are non-significant, usually this means over-fitting.

If you can add more significant coefficients but do not do that, this is under-fitting.

No to the guy above.   Just because a variable is not significant does not mean it is overfitting the model.  Depending on modeling technique you can look at multiple different things.  Since you are doing regression analysis (i'm guessing by the topics):

1. You can look at how the adjusted R^2 changes (is it only a minor change then you are not getting enough out of adding the variable and might question it's inclusion).   This also can be found in the Mean Square Error (MSE) in the anova table output for a regression model.  If that is not changing much then question including a variable.

2.  Mallows's Cp estimates the ratio of overfitting a model to underfitting a model (treats your entire model as the theoretical standard).  

3.  Sequential ANOVA or TUkey's sweeping out method will help you estimate the exact amount of variation in your outcome each variable is accounting for (sweeping out sweeps out shared variance, you can estimate a few sequential ANOVA's to explore shared variation but doesn't remove it).  Either way if a variable performs horrible in the sequential ANOVA then it mAdd your answeray be overfitting the model.

Myers, Raymon H. (1990).  Classical and MOdern Regression with Applications (eds). is a good source for info on some of the model selection (exploring overfitting/underfitting) 

Or if you care about the theoretial background behind Mallow's Cp: 

Mallows, C. L. l973. Some comments on Cp. Technometrics, l5: 66l-75. 

Meghan L. Rogers:  I think "looking at how R^2 changes" is not a formal statistical test. If you want to carry out a test based on how R^2 changes, you still will need to deal with significance of coefficients.

P.S.

@ Meghan L. Rogers: The question relates to modelling a non-linear trend, and there are many publications on that topic based on assessing the significance of coefficients.

For example, there's a simple, efficient & well-known algorithm based on testing the significance of regression coefficients to see how many variables we need for non-linear approximation.

The method is  based on a backward elimination procedure (Smith, 1982).The procedure starts with a large number of equidistant knots (so that there are about four or five data points per one knot). Then the number of knots is reduced by one at atime until all the regression coefficients for the remaining knots become statistically significant.

Papers:

Smith, P. L. (1982). Curve fitting and modeling with splines using statistical variable selection techniques(Report NASA 166034). Langley Research Center, Hampton, VA.

Stone, C. J. (1986). The dimensionality reduction principle for generalized additive models. Annals of Statistics, 14, 590-606

Xu -

A model should make good sense in light of subject matter theory.  If you are doing a time series, jumps should only occur when there has been some hopefully observable change which caused a break in the time series, and perhaps justifies a change in the model.  

If you overfit to the point of 'connecting the dots,' you have completely specified the past, but not the future.  It may appear that you have no variance and no bias, but the model is not justified.  If you underfit, it should not forecast well for new cases either.  Something more reasonable is needed which is justified by subject matter theory, new or old, that might possibly even have more than one curve, but only as justified by subject matter reasoning. 

The variance of the forecasting error may be estimated when forecasting for a point, but bias may not be so readily estimated.  Accuracy could be checked, post hoc, however.  What you need is test data for validation.   For example, you could pretend you do not have parts of your data, and see how well you would have forecasted for them, based only on the other data.  You may wish to research validation methods for trends.  (My work was more in cross-sectional surveys, not time series, so "prediction," not "forecasting."  But I recall a time series, an ARIMA use, as I recall, with an additional variable, and enough overall complexity that it turned out to appear to have been overfit to the first data, as it's performance, as measured against two-period ahead checking, turned out to slowly deteriorate over time.) 

Scatterplot analyses, including residual analysis, perhaps some form of control chart, may often be helpful. 

As for hypothesis tests, they are often misinterpreted, and not of great practical use.  

Cheers - Jim

The post hoc accuracy checking requires specifying the hold-out sample, which is a quite subjective parameter itself. However, generally the approach seems to be attractive.

Looking at theoretical background of the process is not always an option.

The 'Sequential ANOVA' mentioned above is equivalent to using the t-test.

And if we use other tools, such as Cp, AIC(c), PRESS, etc., this is not hypothesis testing.

If we do not limit ourselves to hypothesis testing procedures,  the PRESS statistic can be used:

https://en.wikipedia.org/wiki/PRESS_statistic

It is calculated for a number of candidate model structures for the same dataset, with the lowest values of PRESS indicating the best structures. Models that are over-parameterised (over-fitted) would tend to give small residuals for observations included in the model-fitting but large residuals for observations that are excluded.

It looks like a typo in the linked Wikipedia entry.  If I'm reading it correctly, the "-i" part of a subscript should not be there, unless somehow that author hoped it would help remind one as to what data are used for the forecasted value, but I do not see that it helps, or how it tells one anything meaningful, so maybe it's just me, but you might want to drop that part when you read that.

By the way, as a point of semantics, many may use "predict" and "forecast" synonymously, but that often leads to confusion:

 https://www.researchgate.net/publication/275365705_When_Prediction_is_Not_Time_Series_Forecasting

Research When Prediction is Not Time Series Forecasting

The subscript "-i" seems to be used to show the out-of-sample predicted value calculated for the omitted observation.

Without the subscript "-i" it would just be estimated value based on all available data points.

Andrey. OK.  I did not read it that way, but OK. 

Thank you for all of your reply. These answers are helpful. In the past 10days i'm on Chinese new year vocation. Deeply Apologize for that delay. There is still a question puzzed me is how to  judge when to use a piecewise linear model and a quadratic/cubic model? (We assume there is only one dependent variable and independent variable, seperately)

You can use AIC or PRESS to choose the best approach.

Akima spines can be even better:

http://www.jive.nl/jivewiki/lib/exe/fetch.php?media=expres:fabric:interpolation.pdf