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I have two time-series variables: each has 14 points with an interval of 5 years. The precise years are:

1950、1955、1960、1965、1970、1975、1980、1985、1990、1995、2000、2005、2010、2014. 

I want to do the unit root test and cointegrating test for each variable. But I think the number of observations is too small. So I want to interpolate this 5-year data to annual data. Because my variables are land areas in specific years, and they are increasing year by year as the data are based on large scale and the general trend is growing.

I have tried to interpolate the data by linear interpolation: the time-series plots can be seen as below, the blue points are original points, the red ones are interpolated:

time series plot of protected area

time series plot of urban area

Because the data of one variable (protected area) are based on my spatial analysis, and each layer costs much time for PC to run. So I want to interpolate data to add observations. In other hand, because the protected area increased by a stable rate (no drop or very rapid increase), so I think even after I use spatial analysis processing to generate the data of middle years, the results won't have much discrepancy with the interpolated ones.

I want to know whether can I do this when considering the characteristics of my data and my data processing purpose. And if doing this, anything should I notice when I do time-series processing?

These was a similar question on Cross Validated: however, it hasn't be solved: How to interpolate independent variable over five-year period?

My data are as follows:

Year,Parea,Uarea
1950,3435829.43 ,144179.7476
1955,3619503.16 ,168028.4699
1960,3881482.63 ,196839.0495
1965,4310040.34 ,229032.161
1970,4950230.51 ,262543.7928
1975,6216028.19 ,297502.4439
1980,7062749.74 ,337481.6276
1985,8187770.34 ,381059.4338
1990,9893501.67 ,432255.4666
1995,12011196.93 ,487330.1703
2000,13327189.88 ,546829.7056
2005,15231484.09 ,612606.1358
2010,16986859.05 ,683200.605
2014,18097951.40 ,743693
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    $\begingroup$ Note that I previously commented in reply to your previous question: "Interpolating to get data every year would not be completely crazy, but would not be a good idea unless you took account of that in your modelling. I can't see how to do that, as most of your points would be interpolated, and the gain in degrees of freedom would be essentially spurious." stats.stackexchange.com/questions/126001/… What we tell you repeatedly (see answers here) is true! $\endgroup$ – Nick Cox Dec 5 '14 at 10:26
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    $\begingroup$ The fallacy here can be seen by considering why you don't interpolate to monthly, daily, hourly ... data. The sample size would increase, which is good, right? No; not if it is based on exactly the same information. $\endgroup$ – Nick Cox Dec 5 '14 at 10:30
  • $\begingroup$ It's misleading, indeed incorrect, to say that the previous problem cited (stats.stackexchange.com/questions/97399/…) was not solved. The answer to that question was as here, that the proposal to get more data by interpolation is a bad idea and the increase in sample size is just spurious. $\endgroup$ – Nick Cox Dec 5 '14 at 10:41
  • $\begingroup$ @NickCox, I didn't notice that it was the same OP asking a similar question in May, though I remember answering it. I can only wonder if there's some context missing here. OP mentions some sort of spatial analysis to obtain the data, it may help to know more about the method. $\endgroup$ – Aksakal Dec 5 '14 at 14:26
  • $\begingroup$ @Aksakal It's not the same OP from the evidence we have. $\endgroup$ – Nick Cox Dec 5 '14 at 14:28
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I want to do the unit root test and cointegrating test for each variable. But I think the number of observations are too small. So I want to interpolate this 5-year data to annual data

You can't do this. You're trying to inflate the significance level of your statistical test. If you need a bigger sample, then get a bigger sample. You can't increase the sample size by interpolating.

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There are multiple ways to input data, linear interpolation (as you suggest), splines, or any other complicated way you can dream of. However, regardless of the method you choose, you will be artificially reducing your uncertainty by using the data more than once.

As usual with small sample sizes, this is a scenario where if you can quantify your uncertainty in the form of prior distributions, being Bayesian can really pay off. If you're interested, this book by Raquel Prado and Mike West is a good place to start.

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