Fisher VS Reality: Comparison of Statistical Methods

[] – Part E of Lec 5:Descriptive Statistics: An Islamic Approach. This lecture explains the difference between classical Fisherian approach and our REAL statistics approach, within context of a study of the Quantity Theory of Money.

In previous portions of this lecture, we have emphasized the need for a new approach, which we call “Real Statistics”.  In this lecture, we illustrate the differences between the conventional approach and our new approach using the already studied example of Australian Inflation. In this connection, it is of great importance to understand the following:

The DATA is ALL we have – The STATISTICAL ASSUMPTIONS imposed on the data DO NOT PROVIDE US with additional information. HOWEVER, all statistical inferences we make RELIES HEAVILY on these UNVERIFIABLE (and typically false) ASSUMPTIONS.

First Step of a REAL analysis: LOOK at the DATA with reference to a REAL world issue under examination. In this case, we are interested in the Quantity Theory of Money in general. In particular, we want to examine Milton Friedman’s idea that “Inflation is always and everywhere a monetary phenomenon, in the sense that it is and can be produced only by a more rapid increase in the quantity of money than in output” The data can tell us whether or not this important hypothesis about the economy, which asserts the neutrality of money, is true.

Graph of Prices (GDP Deflator) and Broad Money. Money has been rescaled to be 100 in 2019, just like the price index series. This data is taken from the WDI data set.


The two graphs clearly show different trends. In the early period from1972 to 1990, prices are increasing sharply, while money is increasing slowly. Later, Money starts to increase sharply while price curve is flatter, showing a smaller rate of growth. Looking at the graph leads to the IMMEDIATE conclusion: There is no strong direct relationship between money and prices. Note that this conclusion is based on direct examination of data, without any stochastic assumptions required for the Fisherian approach.

Second Step: Look at DATA in WAYS which are relevant to ISSUE of concern! In this case, the QTM tells us the increases in money stock lead to increases in prices. To examine this, we need to look at the Rate of CHANGE in prices, and also the Rate of CHANGE in money. In previous analysis, we came to the conclusion that the best measure of rate of change is the following:

  • Define %P = log{P(t)/P(t-1)}
  • Define %M = log{M(t)/M(t-1)}

With this definition, the growth rate of money over two years will be sum of the separate growth rate for each of the two years. A Graph of %P and %M is given below:



This graph shows a ROUGH correspondence between the two series, but also shows many anomalies. That is, there appear to be sharp increases and decreases in %P which do not correspond to any similar change in %M. It does not seem that %M can explain all the fluctuations in %P, contrary to Friedman’s dictum. However, this is just a preliminary impression, and more careful analysis is needed to come to firm conclusions.

Third Step: Find ways to analyze the data SUITED to the question you are asking. The jagged graph above contains too much information, and is not directly suited to telling us about: “How STRONG is the ASSOCIATION (not causation) between %M and %P?” Note that association is symmetric, and causation is uni-directional. Even though we are interested in the causality – does %M cause %P or is it the reverse? – the data cannot tell us about this crucial question. Techniques for studying causality are extremely important, but are not part of conventional, Fisherian statistics. In fact, many famous statisticians are on the record as having denied the relevance, importance, or even meaningfulness of the idea of causality. We will not look at this debate in any depth in this current course, which deals with elementary concepts only. However, causality must be an important part of any “REAL” statistics.

As a first step towards the deeper and more complex concept of causation, we can try to measure contemporaneous association. “Contemporaneous” means that we look at the relation between %P(t) and %M(t) for the same year t – we do not look at associations across time. This is always a useful first step. The STANDARD METHOD in use for this purpose is as follows. ASSUME data is jointly normal. Apply the formula for the correlation coefficient; this is the best measure of association for Bivariate Normal Distribution. As with all methods of conventional Fisherian statistics, this method suffers from a serious PROBLEM: it works VERY POORLY if data is not Normal. There is NO REASON to assume data under examination is like a random sample from a hypothetical infinite bivariate normal population. Instead, we develop a direct and intuitive method for evaluating association below.

As a first step, divide %P and %M into HIGH and LOW. We want to know if %P is High when %M is High, and also whether %P is low when %M is Low. The question is: How to divide a series into HIGH and LOW parts? There is a NATURAL and INTUITIVE methodology for doing so. We can SORT the series in ascending order in EXCEL. Find the MIDPOINT. We have an annual series with 59 point of data, so the 30th data point would be in the middle. Series data points BELOW midpoint can be classified as LOW, while data points above midpoint are HIGH. This is natural in the sense that the 29 lowest rates of %P within the 59 points are classified as LOW, and the 30 highest data points within the data set are classified as high.

Using this method of classifying %P and %P into HIGH and LOW values, we can make a chart of Lows & Highs for %M and %P as follows:



Many interesting patterns can be seen from the above graph, which shows the highs and lows for both %P (changes in prices) and %M (changes in broad money). First we note that from 1961 to 1972 rates of money growth (%M) were LOW,  except in the two years 1963 and 1964.  Corresponding to this period, we find that %P was low in 1961-66, High in 1967, Low in 1968, and then consistently High from 1969 to 1990. There were two decades of high inflation in 70’s and 80’s was followed by a low inflation period from  1991-2004. This picture immediately leads to many questions:

Why was there an episode of high inflation in 1967? If Friedman’s hypothesis is true, than it must have been due to previous high rate of increase in money.  Could it be that High %M in 1963, 1964 led to High %P in 1967? Knowing about the mechanisms of money and prices, this seems highly unlikely. Note that this conclusion comes from our general understanding of how the real world works, NOT from the data itself.

A second important question is “Why did %P become Hi over 69-90?”. In connection with Friedman’s hypothesis, it is interesting to note that %M became high much later than these periods of high inflation. Money growth rates became consistently high in the period 79-91. Why did increase in money growth FOLLOW the increases in inflation? According to one theory, monetary policy should accommodate the needs of business. In periods of high inflation, the need for money is high, and so one should print more money. We need to look at the minutes of the Monetary Policy Committee to see what they were doing and why they were doing it. It seems clear that the periods of HIGH inflation from 1969 to 1979 were not preceded by High rates of growth of money (%M). It seems likely that this inflation came from a different source. Again, we need to more carefully at the real world, and try to find other causes of inflation, to explain the patterns we see in the data.

From this preliminary analysis, it is clear that Real Statistics leads us to ask different KINDS of questions. In Fisherian statistics, we start by ASSUMING that the Data are RANDOM draws from hypothetical population. In this case, the data is ONLY a means to discovering the parameters of this IMAGINARY population. All of the sophisticated mathematical machinery of inference and hypothesis testing deals with issues of how we can use the data to learn about the imaginary population from which the data is ASSUMED to be a random sample. In this scenario, if we know the parameters of the imaginary population, we don’t NEED the data!! The parameters provide us with COMPLETE information about the data set!! The Individual Data points DO NOT MATTER and are MEANGINGLESS – they are all random draws and could come out differently next time. This is in dramatic contrast to real statistics: Each data point matters! We do not come to understand data by imagining a hypothetical underlying population. Instead, we understand Data by looking at the Reality which generates the data. Why did rates of money growth %M become high over 79-91? No amount of playing games with this data will lead us to the answers. Instead, we must examine the Australian Economy, and maybe the world economy as well. We must look at methods of money creation, to find sources for the extra money created. One source is the government. Look at bulletin of the Monetary Policy Committee. How were they making decisions about monetary policy. Were they accommodative or forward looking? What were the variables they considered? But additionally, we may study private money creation by financial institutions. This was a oeriod of Financial De-Regulation. Removal of restrictions led increase in loans and creation of money. This  may be reason why %M was high in this period.

On more technical note, we can also use this partitioning of data to create a simple measure of association, which does not rely on unverifiable assumptions about imaginary populations. We can simply DIVIDE the data into two halves – those with HI %M and Lo %M. Now look at the behavior of %P on each of these halves separately. If nearly all of the HIGH value of %P occur within the High %M data set, then the two series must be highly correlated. Doing this simple counting of the data gives us the following table of counts:

%P vs %M %M=LO %M=HI
Lo %P 20 9
Hi %P 9 21
Total 29 30

We see that in 29 years where the money growth was in the bottom half (%M is Lo), %P is also Low for 20 Years and High in 9 Years. Similarly, for the 30 years where %M is High, %P is high in 21 years and low in 9. This shows a strong association between the highs and lows of %M and %P. When one is high, the other one is also high in roughly 2/3 of the cases. One high and the other low occurs about 1/3 of the time. If there was no relationship between highs and lows of %M and %P, we would expect to see about 50% of the total or 15 cases out of 60, in each of the four boxes in the diagram above. The numbers show a strong but loose relationship. It seems clear that Friedman’s hypothesis, the %M is the ONLY source of inflation, is not correct. While %M does exert a strong influence on %P, it seems likely that there are other causes of inflation as well – about 1/3 of the cases of Hi and Lo %P cannot be explained by Hi and Lo %M in the same period.


There is strong relationship between %M and %P. Direction of causation is not clear, and CANNOT be learnt from the data. INSTEAD, we can learn about causes by “expending shoe leather.” Expending shoe leather is a metaphor for exporing the real world and searching for causes. In this particular case, we must analyze bank statements and Monetary Policy statements, and look at real world factors for money creation and inflation.

We learn that the direct analysis of data, WITHOUT ANY assumptions about stochastic structure, gives us a lot of information about the real world. It is important to understand that  NO MORE INFORMATION is available. Stochastic assumptions REDUCE importance of data, by mis-directing our attention from the data itself to a hypothetical imaginary infinite population, from which the data is assumed to be a random sample.

The lessons of this lecture are summarized in the picture below:


Related Materials: Previous Portions of Lecture 5:

  1. Malthus and the Birth of Statistics
  2. Galton: Eugenicist Founder of Statistics
  3. Fisher’s Failures & the Foundations of Statistics
  4. Real Statistics

An organized version of all the lectures, together with supplementary materials and quizzes, is available as an online course. Currently, it is being offered free, in expectation of receipt of useful feedback for finalizing the course. You can work through the lectures at your own pace, and also ask questions on any of the materials. To register for the course, use the following link: Al-Nafi Portal: Descriptive Statistics (Registration)


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About Asad Zaman

BS Math MIT (1974), Ph.D. Econ Stanford (1978)] has taught at leading universities like Columbia, U. Penn., Johns Hopkins and Cal. Tech. Currently he is Vice Chancellor of Pakistan Institute of Development Economics. His textbook Statistical Foundations of Econometric Techniques (Academic Press, NY, 1996) is widely used in advanced graduate courses. His research on Islamic economics is widely cited, and has been highly influential in shaping the field. His publications in top ranked journals like Annals of Statistics, Journal of Econometrics, Econometric Theory, Journal of Labor Economics, etc. have more than a thousand citations as per Google Scholar.

4 thoughts on “Fisher VS Reality: Comparison of Statistical Methods

  1. Reblogged this on WEA Pedagogy Blog and commented:

    Post shows how conventional statistics attempts to make inferences about imaginary parameters. It proposes an alternative. Real Statistics should use data to make inferences about the real world, instead of an imaginary world.

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  3. Pingback: Do the Rich Have Fewer Children? | An Islamic WorldView

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