Let the data generating process be as follows:
This implies that the price of a car is determined by its milage and its age. However, for whatever reason you omit the variable age in your regression model and you estimate the following reduced regression model:
What are the effects of omitting the variable age? What will happen to the coefficient of miles? What sign do you expect for ?
Let’s elaborate on these questions. A priori, one would expect that a higher milage lowers the price of a car. Hence, we would expect to have a negative sign, i.e. . One would further expect that an older car is cheaper and hence traded at a lower price. Also, one would expect that an older car has more miles. We can therefore conclude that,
What does this imply for our regression analysis? We know now that a large number of miles lowers the price of a car. But, if a car has many miles it tends to be older. Thus, when omitting the variable age, the variable miles may actually be accounting also for the effects of age and not only miles.
Thus, , suffers from a bias.
But can we say something more about the bias? Yes. We know that suffers from a downward bias. This is because both age and miles have a negative effect on the price. Leaving out age lets the coefficient of miles pick up parts of the negative effects of age.
Hence, it follows that the true . This implies that if then it is not necessarily true that .
The illustration below summarizes the direction of the omitted variable bias. Let Y be the dependent variable, A and B the independent variables, and B the omitted variable.
The Venn Diagram below illustrates the problems that arise when we neglect an important variable from our regression analysis. Note that, the overlap of miles and price (area C) is the true impact of variable miles on price. The overlap of age and price (area D) is the true impact of variable age on price. Now, assume that you include millage in your regression analysis, but you omitted age. By doing so, you are estimating the impact of miles on price by areas C and B and not just area C. What can you say about the estimate of miles in the regression? What will be the consequences of neglecting age? Here are some general statements on what will happen if you neglect an important variable, in our case age.
The following post further explains the nature of the omitted variable bias. Particularly, the post discusses the effects of the omitted variable bias on single coefficients.
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The omitted variable bias occurs because of a misspecification of the linear regression model. The problem can arise for various reasons, either because the effect of the omitted variable on the dependent variable is unknown or because a variable is simply not available. In the latter case, you might be forces to omit that variable from your model. However, one needs to be aware that omitting a variable might lead to an over-estimation (upward bias) or under-estimation (downward bias) of the coefficient of one or more explanatory variables.
In order for the omitted variable to bias your coefficients, two requirements must be fulfilled:
In our example, the age of the car is negatively correlated with the price of the car and positively correlated with the cars milage. Hence, omitting age in your regression results in an omitted variable bias.
Part three of the series on the omitted variable bias, intends to increase the readers understanding of the bias.
Generally, one can use various libraries to read Excel files, including XLSXReader, ExcelReaders or Taro. This tutorial will focus on Taro as it created the fewest problems and provides – at least in my eyes – an easy to understand syntax. In order to download and read an Excel file into Julia it is sufficient to execute the following lines of code. This script downloads some sample data provided by this blog and reads it into Julia.
# in case you have not installed to Pkg yet Pkg.add("Taro") # load Taro - Pkg to read Excel Data using Taro Taro.init() # get data path = "https://economictheoryblog.files.wordpress.com/2016/08/data.xlsx" df = Taro.readxl(download(path), "data", "A1:C357") # # The dataframe df contains the downloaded data #
This routine will read the selected data fields, i.e. field A1 to field C357, of data into a dataframe in Julia. You can also directly consult the documentation of the Taro Package here.
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The following code produces the Venn diagram used in the post explaining the omitted variable bias.
# start with an empty workspace rm(list = ls()) # load EulerR pkg library(eulerr) # create sets fit <- euler(c(Price = 500, Miles = 500, Age= 500, "Price&Miles" = 200, "Price&Age" = 200, "Miles&Age" = 200)) png(filename="venn.png") plot(fit, fill_opacity = 0.3) dev.off()]]>
# in case you have not installed to Pkg yet Pkg.add("Taro") # load Taro - Pkg to read Excel Data using Taro Taro.init() # get data df = Taro.readxl("path to Excel", "sheet1", "A1:C357")
This code will read the selected data fields, i.e. field A1 to field C357, of sheet1 into a dataframe in Julia. You can also directly consult the documentation of the Taro Package here.
]]>In contrast to the neoclassical investment theory, that explains investment in the context of an optimal capital stock, q-investment theory explains fluctuations in investment with changes in the stock market. That is, the q-investment theory explains investment using the difference between the stock market valuation of firms real assets and the replacement costs of these assets. According to Tobin’s q-investment theory, firms base their investment decisions on q, where q represents the ratio between the market value of all physical capital and its replacement costs.
In the case that q is above one (q>1), the stock market values the firm more than the market value of its real assets. In this case a firm can increase its value by acquiring additional capital. Thus, firms will investment and increase their capital stock. In the opposite case, when q is smaller than one (q<1), the market value of the firm is less than the replacement costs of the firms assets. It this case a firm does not replace depreciated capital, because the market values the additional investment less its costs. Hence, the capital stock will decrease.
The q-theory of investment implicitly considers the marginal costs of adjusting the capital stock. Hence, in contrary to the neoclassical theory of investment, the q-theory of investment is not primarily based on the assumption of an optimal capital stock, but emphasizes the optimal adjustment path towards the new capital stock.
Suggested Reading:
Brainard, W. und J. Tobin (1968), „Pitfalls in Financial Model-Building”, American Economic Review 58 (2), 99–122.
Keynes, J. M. (2016). General theory of employment, interest and money. Atlantic Publishers & Dist.
Tobin, J. (1969), „A General Equilibrium Approach to Monetary Theory”, Journal of Money, Credit and Banking 1, 15–29.
]]>One can separate investment into replacement and net-investment. While replacement investment serve the purpose to replace depreciated capital, net-investment define actually changes to the capital stock. Net-investment are essential for the long-term determination of the capital stock. The neoclassical investment theory assumes that firms invest if their current capital stock is smaller than the optimal capital stock. Vice versa, firms disinvest in case their current capital stock is sufficiently larger than the optimal capital stock.
The optimal capital stock can change for various reasons. Changes in demand can lead to changes in the optimal capital stock. An increase in demand implies that firms can increase the amount of goods they can sell and attain higher profits. However, in order to be able to produce more goods firms need to expand their production capabilities. Hence, firms need to invest in more capital. Furthermore, changes in financial costs can lead to changes in the optimal capital stock. For instance, a decrease in interest rates might lead to a decrease in financial costs. An investment project that has not be been profitable with higher financial costs might suddenly become profitable and hence increase the optimal capital stock. Nevertheless, the current capital stock of a firm may also change for reasons unrelated to the business cycle. For instance, delays in delivery may defer adjustments of the capital stock.
Besides the neoclassical investment theory, Tobin’s q-investment theory represents a popular alternative theory of investment. The q-theory of investment implicitly considers the marginal costs of adjusting the capital stock. Hence, in contrary to the neoclassical theory of investment, the q-theory of investment is not primarily based on the assumption of an optimal capital stock, but emphasizes the optimal adjustment path towards the new capital stock.
Suggest Readings:
Jorgenson, D. (1963), „Capital Theory and Investment Behaviour”, American Economic Review 53, 247–259.
Jorgenson, D. (1967). The theory of investment behavior. In Determinants of investment behavior, NBER, 129-175.
Jorgenson, D. (1971), „Econometric Studies of Investment Behavior: A Survey”, Journal of Economic Literature 9, 1111–1147.
]]>Consulting the FAQ on the WordAds page (https://wordads.co/faq) does not tell you much: A site generally needs thousands of pageviews each month to earn meaningful revenue, it says. Certainly, this answer does not help. Fortunately, the web is full posts and comments of people speculating about the minimum traffic requirement. Needless to say that none of them truly knows the exact number. We can nonetheless learn from past observations and try to approximate the current number with some estimates. It appears that the minimum traffic requirement of WordAds is not constant over time. When Automattic launched WordAds on November 29, 2011, some blogs were accepted with less than 5000 pageviews per month. However, the number of views required to be accepted to the advertisement program quickly rose after the initial launch. In the following years some bloggers reported that they got accepted when reaching 15000 views per month, others speculated that the number was further rising. Lately, reports on blogs being accepted to WordAds have been rare.
I applied to the WordAds program in January, 2015 with 471(!) monthly views. Clearly, WordAds did not accept my blog right away. So I had to wait. As I am a somewhat lazy blogger, my views did not rise sharply. However, they kept on increasing on a slow, but steady pace. I had some ups and downs until finally in October 2017 I hit the magic barrier: 15000 views per month.
Some month earlier, I read some comments on the existence of blogs that were in the program and had allegedly 10000 views or less. So when reaching 15000 pageviews and still no news from WordAds I got impatient. I finally found the motivation to write WordAds directly. I got an immediate standardized message saying that they have received my mail.
After sending the mail I was expecting a standardized reply forwarding me to the there FAQs. To my amazement, Kris, a happiness engineer @ wordpress, wrote back to me within a couple of hours. And Kris lived up to his job description, I got a positive answer. My blog was approved to the WordAds program with little over 15000 pageviews a month.
We can conclude that the minimum requirement to be accepted at the WordAds program is still around 15000 pageviews per month, maybe even less. To conclude, one thing I noticed when searching the web for hints and reports on the minimum traffic requirement, various bloggers got accepted to WordAds after having wrote them an email. I do not know if there is a causality behind this pattern or if sending an email just triggers the responsible person at WordAds to review your application in more timely fashion. Nevertheless, if you feel you might be in the range of being accepted, it certainly does not hurt to write an email.
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Most DGE models include a number of core of elements:
Households represent the demand side of the economy. We need to specify their preferences over commodities and endowments. We also need to state the households’ objective, which usually is utility maximization. That is households try to maximize their preferences subject to a set of constraints.
Beside households, we need to specify the supply side. We specify an amount of firms and a technology that firms can use to produce and sell goods. Furthermore, we also need to define the goal of firms. Usually, the objective of firms consists in maximizing their profits subject to their production plans being technologically feasible.
Additionally, we need to specify institutions, which usually includes a government and/or a central bank. Particularly, we need to specify the policy of the institutions. Thereby, the policy of institutions can be taken as given or being modelled as optimally chosen, subject to a government budget constraint.
It is important to specify the information in the economy. That is, we need to specify who knows what at which point in time.
We need to implement markets and think of what kind of market we wish to model. Popular choices of markets include Arrow-Debreu, Sequential Markets and Incomplete Markets.
Finally, we need to specify the equilibrium concept. A possible choice is the competitive equilibrium, in which the price cannot be influenced by a single firm.