Have you any reasonable arguments about the long term behavior? I mean, does your 'situation' under examination perceive the well known concavity in the long term or is it something different? Any time you have to think about the long term: in the short term we are linear, in the long term usually we are not.

I think cross-validation is one the best methods to compare different types of models, linear and/or nonlinear. You may skip some observations (K fold cross-validation) and construct your models using the rest of the data, then predict these neglected observations. Repeating this procedure till you cover the whole data set, the model will the smaller PRESS is deemed a better model.

To Demetris: Its a timeseries data for 63 years. Concativty is visible, but the curve is broken at few points. I fitted piecewise linear function and exponential function, and just trying to choose one among.

To Balazs: The function what i should decide is the starting point. As told to Demetris, a piecewise linear function and an exponential function, both seems well to fit for the series, observing thorough scatter. Trying for a best among them. I don't have much idea about cross-validation techniques.

I guess your regressor (or sometimes named "independent variables", but actually misnomer) variables in the two models are different.

In its SIMPLEST way, you may use cross validation in this manner: Randomly divide your data into two, and fit your model1 and model2 to the first half of the data. What you obtain would be the parameter estimates which would minimize the error for the first half of the data. Now using these parameter estimates and regressor data (I guess "time" values) for the second half, estimate the rice productivity for this second half. Find the squared error for the second (estimated) part only, and record this value. Now change the order, use the second half of the data for parameter estimation and the first half for productivity estimation. Find the squared error for the first half. Add these two squared values. This would give you prediction error sum of squares (PRESS) value for each model (sometimes this value is divided by number of observations, but this would not change the result). The model with the smallest PRESS residuals is the best one.

However, this two-fold CV is not highly recommended, my experience with CV has shown me that if you have a large data set, then dividing the whole data into at least 8 subgroups would be more appropriate.

To Balázs Kotosz: If the dependent variables differ (which, I guess, is not the case here), then normalization can be used for prediction error terms of the two response variables. Of course the exact method would depend on the application (and the units and penalty functions, etc.), but a very straightforward normalization function would be to scale the prediction error terms via dividing by the total variance of the response variables. Or maybe first auto-scaling both response variables then applying cross-validation would also work, I guess.

Dear S.J. Balaji, as I can understand and because of the limitations in food production, probably the better model could be a sigmoidal one. A sigmoid function is increasing exponentially untill an inflection point, then is increasing with a diminishing rate and finally takes an horizontal line shape. This curve at small time intervals is close to straight line, but piecewise linear is not a good method to fit because of the gaps between the components. Read more about sigmoid function here:

http://en.wikipedia.org/wiki/Sigmoid_function

and figures here:

https://www.google.com/search?q=sigmoid+curve&espv=210&es_sm=93&source=lnms&tbm=isch&sa=X&ei=E8bcUuHMEeiZyQPUsYGABA&ved=0CAkQ_AUoAQ&biw=1599&bih=809

Hi all, the dependent is variable is rice-productivity and the regressor is time variable. Assuming linear & exponential forms, the dependent variable takes the forms Y & ln(Y), where ln(Y) appears for linearizing. So, in 2 models, regressor is same, the time element. I have no problem in linear model. In exponential function, as Y is not equal to ln(Y), neither i can use R-Square / Adjusted R-Square. AIC / BIC criterion in "much" less for exponential model, suggesting in favor.

Shall i do like this? From estimated parameters in exponential model, i will find the predicted values of ln(Y). Then i can find out actual Y(fitted) values by taking 'antilog' and can obtain residuals. From the residuals, i can find RMSE (Root Mean Square Error). I can obtain RMSE from the 1st linear model. Would it be reasonable to compare RMSE of these 2 models and would decide the model having minimum RMSE?

To Demetris: Thanks. The scatter line for rice productivity doesn't show a sigmoid shape in my case, may be the actual curve is part of the whole sigmoid function. I will try this too in these days.

To Burak & Balazs: Thanks. I'm new to the cross validation and learning now. Is dividing the whole data set into "two" is sufficient, or i could do it for more segments. I was referring a bit on k-fold and v-fold cross validation methods, but actually i can't find actual difference b/w k-fold & v-fold. Kindly give an idea on this.

Not sure if this is useful to you but I found it interesting:

Automatic Selection for Non-linear Models

http://ideas.repec.org/p/oxf/wpaper/473.html

Let's generalize and think to the concept of model.

What's a model? Is an approximation of the reality.

So when two model are more or less comparable, to decide which is the best one you have to put in your decision your knowledge about the problem.

Do not think only in statistical terms but find a rationale.

The question is: do the rice productivity in time should be linear? what's your expetation about that? At mid-latitudes for example I could expect for a seasonalized shape of the curve, while at tropics not. What's the hint coming from observing the scatter-plot?

Think simple.

I think look at R^2. The best that gives high R^2.

Hi there;

if you are trying to model the rice productivity in function of time, then your dependent variable for both models is "rice productivity". I do not understand how the dependent variable becomes different for both models!

Back to basics: before running a model you should plot the scatter plot of the data, I guarantee you that if the scatter plot has a linear trend; the linear model will perform better than any no-linear model.

My advice is that if you have two different dependent variables, you should use a vector autoregressive regression model (VAR) especially when time is important as it is the case for your exercise!

Good luck!

DR John MM

I think Mr Amro is right, look at R^2 (goodness of fit test). The higher the R^@ gives best model.

You will look at the R^2 of the two regressions if and only if the dependent variable is the same for both models.

Compare apples with apples!!!

Dr John MM

To John: OLS won't work for exponential function, as parameters to be estimated are non-linear. Once i take natural logarithm on both sides, now the right hand side is linear, and now, OLS works. But now Dependent variables is not "rice productivity", it is "natural log of Rice productivity". Now i cant compare a simple linear model with this model, via R-Square / Adj. R-Square, bcoz the dependent variables are different. But the problem is i should choose one among them......and thats the block i face...

Without seeing the original data it is difficult to make any recommendation. In any case, a fitted linear trend will provide a constant growth rate during the whole historical period. Is this reasonable or supported by the data? If not, then you have to go back to the original specification, no matter how good is the statistical fitting of your model.

You could try an approach that is conceptually nested, suppose r is the growth rate and R0 is the amount of rice at TIME=0 and MAX is the maximum rice production possible:

1. linear growth: RICE(t) = R0 + r • TIME

2. exponential growth: RICE(t) = R0 • e^-(r • TIME)

3. restricted growth: RICE(t) = MAX - (MAX - R0) • e^-(r • MAX • TIME)

4. logistic growth: RICE(t) = (MAX • R0) / ( R0 + (MAX - R0) • e^-(r • MAX • TIME) )

e^(...) = exp(...) in most software packages

Comparison of information indices / deviance should give you an idea what kind of (idealised) growth curve you are dealing with.

There is a philosophical discussion in this direction, viz., which one to choose when two models are equally performing well with the given data. "Occam's Rajor" says, select the simple one, because often simple solutions are more general in their usage. You may please refer to "structural risk minimization" and "regularization" theory which extensively deals with this direction of research.

Thank you and best wishes.

If you can write the linear model as a restricted form of the exponential one (for example, if several parameters of the exponential model are restricted to 0, then you have the linear model, you haver several restriction tests, for example, the likelyhood ratio test.

In case you cannot explain one of the models as a restricted version of the other, you can compare the Akaike information criterion for both models. The model with a LOWER value for the Akaike Information Criterion is the best of them.

The underlying science should provide a clue to which model seems to match the underlying science. If this does not yield a clue, there are methods of 'fidning' the best fit. You may read http://en.wikipedia.org/wiki/Curve_fitting.

This problem is very common in structral and system identification in engineeringm and in statistics.

When you use a nonlinar model the R^2 can not be used.

You can check the residual properties.

I agree with Mr. Pulabaigari, and I think you should go with a semi-parametric test, I suggest you Akaike Information Criterion.

If you have the model like in Hasselman you can use R2 after the transformation but you have to consider the intercept necessarily (e.g.RICE(t) = alpha +R0 + r • TIME versus RICE(t) = alpha+ R0 • e^-(r • TIME) ) otherwise you can have no measure for comparisons. R2 is available in statistics only if you have the intercept in the model.

Check the link if you are interested in more complex autoregressive models.

http://theglobaljournals.com/ijsr/file.php?val=MTQ1MQ==

You should definitively verify common assumptions, especially regarding residuals and time dependency. without verifying those assumption you cannot use R^2 and later you also can use simple metrics like MSE, MAPE for the fit as well. Finally, you should consider the more parsimonious model as recommended in multiple literature about modeling.

I would apply dominance and encompassing theory, see the papers of Hendry, Richard and Mizon. You can apply to non-nested models but can't remember if you can do for non-linear models, it is worth checking, though.

Good luck!

Akaike's information criteria (AIC) or AICc is one way to quantitatively compare non-nested models, and it doesn't matter if one is linear and the other is not. Note that if you apply this approach (over say null hypothesis significance testing, NHST, e,g. via the likelihood ratio test, if you can get the models to be nested) is that it is strongly recommended against to then switch to NHST and use e.g. the standard p-value printout that often comes with fitting logistic regression which are often (always?) based on non-information theoretic arguments. Burnham and Anderson's text on multimode inference is a pretty tractable text on these topics.

Plot your data. Plot your residuals. Consider the problem posed by Anscombe's quartet (http://taggedwiki.zubiaga.org/new_content/1ecb006e117aa06fd4635536e73961ab)

Split the data into two parts, fit both models to one part of the data and then Forecast

the second part of the data sample and run the usual forecast versus actual tests. If you try nesting and testing see Pagan's work on this. There are easier built in tests for nested models in EViews.. And re read alll of the above judicious comments of others all of whom have some good nuggets of wisdom.

When in doubt always choose the model with the fewest parameters.

I would use Log likelihood ratio tests to determine which fits better. However you state that they both are ok, so the question is does the more complex model add anything?

Also what is the theretical model ?

Simply use the LINEAR model as its the simpliest model and theory favors the linear fit unless there is a theretical model to suggest an Exponential model.

I also recommend the book "Modelling Nonlinear Economic Time Series" by Timo Terasvirta, Dag Tjostheim, and Clive Granger

A simple solution is to split your data in two parts: past training and recent testing data. The best model trained on the train set is the one which generalizes best on test set. This is a basic way to avoid over fit.

i forgot to mention the easiest test, the Box Cox transformation. See http://staff.ustc.edu.cn/~zwp/teach/Reg/Boxcox.pdf

Tanguy Urvoy is right. That is called cross validation.

In addition, one should try linear curve fitting, and then quadratic curve fitting, and cubic curve fitting, ..., and choose the best fit (a certain smallest total error). If the basis functions are not polynomials, for example in a Kriging model, there is a power term too.

In polynomial chaso expansion, usually no polynomilas higher than 3 are used for multi-variate systems, due to cost issue.

I would suggest to check first the presence of non linear dynamics using surrogate data approach. Based on the result of the test you can decide to apply linear or non linear model. Initial papers on surrogate data approach are Theiler J et al, Physica D 58: 77-94, 1992 and Prichard D and Theiler J, Phys Rev Lett, 73: 951-954, 1994.

Hi all, Wish to thank you all for detailed discussion.

With Viswanath's quote, "Occum's Rajor" wins. It is possible to obtain R-Square and RMSE of the original exponential function from of linearized (through natural logarithmic transformation) by substituting the anti-log of fitted values from linearized to actual data.

Thanks to Edwards, the BOX-COX confirms same. And last, linear proves to be best.

I always transform the data so the dependent variables are in the same form.Don't forget that you have to use the determinant of the Jacobian when making the transformation.

Dear Dr Balaji

Yes, occam's razor is often helpful i.e. what is the single simplest model which fits the data. In addition to the suggestions made before. However you may like to consider this, in relation to modelling rice production over time, and that is to consider carefully what are the factors that influence the rice production and hence what are the biologically or economically most relevant and plausible model fits and then use that. Best wishes

A simple approach of checking F-statistics of the models under observation could be followed. the one with the highest F-statistics value will be the best fit

Hi Avijit, Thanks for suggestion. I was aware of those factors, of course, whose nature/trend can't explain exactly the productivity behavior. I mean, factors are many and much are inter-dependent (endogenous). And all factors don't exhibit similar trends. When some move up, other, either remain constant or in other combinations. That is why simply i wanted to fit it simply on time........

I think you should consider theoretical factors on deciding which model you should select.

If the propose is Prediction the simpler model is the prefer one.

You may conceal small part of your data, estimating the model using the remaining data, then see if your predictions for the conceal data is satisfactory. One may use the simultaneous prediction region to see if they include the predicted response.

i think that you should adopt the strategy of Terasvirta (1994) of choosing between the linear and non-linear model (Smooth Transition Autoregressive; STAR model), if you use the Autoregressive model.

I completely agree with Rami, from what I understand, you have a time series data for rice-productivity. Is this monthly data or annual data? If it is annual, you don't need to worry about seasonality; if monthly data, you could use 11 dummy variables for months as predictors in your Series Regression analysis. From your question, the trend in time can be fitted by a either a linear or an exponential model - try both, compute R-square for both and use the one with larger R-square (adjusted). If the residuals from this model are autocorrelated, you fit a Time Series Regression model and verify that the residuals from this 'final' model are not autocorrelated.

I am not sure your comment about not comparing two models with different predictors applies in this situation where you are using different trend functions.