Predicting wheat yields

I want to get a function that predicts wheat yields on the basis of climate and soil. I have extracted (with Malcolm help at the GIS end) some basic climate data from an old map of Britain. I did a regression of wheat yields against elevation, amounts of rainfall and length of the growing season. The results are below. This is what the results mean: ELEV is negative, meaning the higher up the farm, the less the yield. Makes sense. GS is growing season....strangely the longer the growing season, the lower the yield. The two 'RAINFALL' variables show the effect of rainfall of 1000mm a year and 1250mm a year. The coefficient for the 1250 is greater (at -4.42) than the one for 1000 (at -1.34). Meaning is that the greater the rainfall, a large reduction in yield. See the adjusted R-squared on the right? That gives us the percentage of the variation in wheat yield explained by the model. Here it is 0.3094 or a tad over 30%. This isn't very good, but better than I had expected. Now I need to add soil data and more accurate climate data, such as hours of sunshine.

Market distance and wheat yields

I am still having problems sleeping because I can't understand why we have some positive signs for distance to market. The theory is that the sign should be negative, meaning that the further the farm is from the market, the lower the rent. Makes sense, doesn't it? So why am I getting a positive sign for some regions in the the south-west of England. Could be related to relatively high yields. Below is a map showing in the top panel our 715 parishes with the wheat yields. In the bottom panel is a map showing the magnitude of the coefficient of the variable 'market distance'. The areas I have circled in the top panel show high yields, in the bottom panel a positive sign. The two areas seem to correspond, don't they?

So it could be that the gains to the farmer of high yields more than make up for distance to market. I'll work on the math to get a function for this and then test it.

Climatic data found

Malcolm has scored what look's like a bulls-eye. I asked him to locate historical data that we could use to test my hypothesis about weather causing a positive sign in the distance to market regression. Today he found some weather maps which look just the thing---I am particularly interested in July rainfall and August/September sunshine. Wheat does best with good rain in July and then a hot dry couple of months. Somewhere also in his find is some historical time-series data. I need that to calculate the variance. The weather maps give an average which is really useful, but it would be nice to know the variance. The two sequences {1,2,3,4,5} and {3,3,3,3,3} have the same mean but very different variances. If you were a farmer, the historical variance in your rainfall and sunshine might alter your cropping pattern and therefore the rent that you might bid for the use of the land.

Possible interesting solution to distance to market mystery

I haven't been able to sleep these last two nights for thinking about why distance to market might increase rent (ie has a positive sign in the regression). I posted something about this mystery a few days ago. I think the solution might be connected with perceptions of risk. Farmers who face stable long-term conditions with regard to climate can generally out-bid farmers who are concerned only about the short-term. Folks with deep pockets can wait out the troughs because they aren't so worried about bringing home food to their family every single day. They can store food or buy it. So it is possible that the unusual pattern of a positive sign might be caused by highly variable weather conditions in that location. So I need to go back over metereological records and calculate the coefficients of variation for temperature and rainfall in various parts of the southwest. I'll put the numbers into the regression and see what happens. This is fun!

Track, rent and causality

I have been working on the mathematical model for the relationship between the laying of railway track and changes in rent. A bit of a problem is showing that there is 'causality' between the two. How can we prove that track caused change in rent? The answer is we can't, and the whole area of causality is frankly speaking a philosophical minefield. It is extraordinarily difficult to show that one thing 'causes' another. For our purposes, the only tool we can use is Granger causality which tests the relationship using time. What we want to see is that rent changes AFTER a change in track, not simultaneously or even worse, before. Malcolm has just given me the track for our eighth estate and I have done the Granger test on all of the them. I'm pleased to say that they all scraped through, some only just. I limited the range of years from 1832-1882 which gives us a half-century. We don't have thorough track measurements for the period after 1872, and wheat and cattle prices were highly volatile in the 1880s. That's OK...we've made the point.

I've been using an interesting form of regression, called Vector Auto-Regression or VAR to get out the stats.  It is simply beautiful! Life doesn't get better than this!

Wheat flows within Britain

Mi has helped me with the data for wheat flows within Britain towards the end of the 19th century. The map shows whether a county was in surplus or deficit. The calculations are for production by county minus consumption within the county. What is left over could be carted and sold to another county. As you can see, the pattern is predictable: the counties in blue had a surplus. These counties are on the arable lands towards the east and south of the country. The areas most in deficit were the sheep raising and also industrialising counties, in red and orange. The country as a whole had a net deficit which was covered by imports from Ireland and also Prussia, and later on, the United States. The point of the map is to show that not much grain moved within Britain by rail. It went by coastal steamer.

Calculating net flows is a useful step in analysing the agricultural structure of a country. We could if we wanted expand the scope to include European countries and North America. Here we had to make an assumption that the per capita consumption of wheat (in the form of bread) was the same across counties. There is evidence for and against that assumption.

Malcolm's comment about railway shareholding

Malcolm made an interesting comment on my last post. We were discussing whether the fact that the owner of a large estate invested in railway shares was interesting or not. Malcolm pointed out that the very substantial investment by the Earl of Leicester, owner of the very large Holkham Hall estate, in railway shares, was just prior to the expansion of railway track near his estate. This would indeed be interesting if:

• we could see a pattern, such as other estate owners also making large investments
• we could infer from this that the estate owners knew that they would be able to increase rents as a result of 'extracting' the savings from their tenants. If thus was the case, the estate owners were getting a free ride: their investment in railway shares would be (probably) be profitable AND they trousered the extra rents. Nice work if you can get it!

The difficulty with this is finding out whether an estate owner was a shareholder. Records do exist, but it would be a monumental task to go through them. Unless anyone has seen a searchable database?