tidy the headings

Signed-off-by: Nathaniel <NathanielF@users.noreply.github.com>

Author: NathanielF

docs/source/notebooks/iv_weak_instruments.ipynb

@@ -55,9 +55,7 @@
     "\n",
     "The factors which determine success in schooling and subsequent employment are diverse. There are many influencing factors at play in your social demographics and individual ability. Consequently there are legitimate concerns about how the effects schooling can be cleanly assessed. Without trying to account for the confounding relationships between ability and schooling we risk biasing our results. In this example we will look into the infamous econometric question of financial returns accruing to investment in schooling. What is the expected increase in my lifetime wealth as a function of my educational attainment? The suggestion from the econometric literature is to leverage instrumental variable regression to decompose the effect of schooling on future wage acquisition. \n",
     "\n",
-    "In particular the suggestion is that we can use an individual's (assumed as-if random) proximity to a 4-year college as an instrument. Recall that this instrument `nearcollege` has to be: (i) __relevent__,  in that it effects the causal path of interest interest i.e. nearness to a college is assumed to have a causal impact on an individual's `education`, (ii) __independent__, such that assignment to instrument be as good as random thereby mitigating the original selection effect bias and (iii) needs to meet the __exclusion restriction__ i.e. the instrument should influence the outcome `wage` only through the treatment of interest `education`. If these conditions are met then, we can argue that the instrumental variable regression techniques can distil the causal impact of our treatment. \n",
-    "\n",
-    "In many ways instrumental variable regression is a paradigmatic method of the [credibility revolution in econometrics](https://en.wikipedia.org/wiki/Credibility_revolution#:~:text=In%20economics%2C%20the%20credibility%20revolution,experimental%20and%20quasi%20experimental%20methods.) and the social sciences. Below we will elaborate how to explore the assumptions of the IV design and highlight the conditions under which it can be sensibly applied. "
+    "In particular the suggestion is that we can use an individual's (assumed as-if random) proximity to a 4-year college as an instrument. Recall that this instrument `nearcollege` has to be: (i) __relevent__,  in that it effects the causal path of interest interest i.e. nearness to a college is assumed to have a causal impact on an individual's `education`, (ii) __independent__, such that assignment to instrument be as good as random thereby mitigating the original selection effect bias and (iii) needs to meet the __exclusion restriction__ i.e. the instrument should influence the outcome `wage` only through the treatment of interest `education`. If these conditions are met then, we can argue that the instrumental variable regression techniques can distil the causal impact of our treatment. "
    ]
   },
   {
@@ -124,6 +122,14 @@
     "pgm.render();"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In many ways instrumental variable regression is a paradigmatic method of the [credibility revolution in econometrics](https://en.wikipedia.org/wiki/Credibility_revolution#:~:text=In%20economics%2C%20the%20credibility%20revolution,experimental%20and%20quasi%20experimental%20methods.) and the social sciences. Below we will elaborate how to explore the assumptions of the IV design and highlight the conditions under which it can be sensibly applied. \n",
+    "First we will outline the nature of the causal estimand in IV regressions. Then we shall unpack how these estimation routines work with respect to the above assumptions. We will explore how different ways of interrogating the models and comparing among similar models helps argue for credible results in these IV designs. "
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -148,7 +154,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Digression: Sampling Multivariate Normals\n",
+    "#### Digression: Sampling Multivariate Normals\n",
     "\n",
     "How can we measure this correlation between instrument and treatment? How much correlation should we expect? In the `CausalPy` implementation of instrumental variable regression we model this correlation explicity using an LKJ Cholesky prior on a Multivariate Normal distribution. It's worth a small digression here to show how sampling from this distribution under different priors can shape the correlations of the joint-distribution. We'll show below how this offers us a mechanism to impose constraints on our beliefs about the relationships between our instruments. "
    ]
@@ -1155,7 +1161,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### The Reduced Form\n",
+    "#### The Reduced Form\n",
     "\n",
     "While in the case of __relevance__ we can aim to justify our assumptions as above, but with the __exclusion restriction__ assumption we cannot directly test the claim. Remember we're arguing that the influence of our instrument $Z$ impacts our outcome $Y$ only through the treatment $T$. In our case this means that the exclusion restriction assumption would be violated if proximity to college had an impact on earnings independent of the mediating effect of schooling. This is presumably not impossible. For `nearcollege` to impact `wage` through a mechanism not related to `education` one would have to posit that there is some driver of wage growth or degradation influenced by proximity to education. You might argue that the density of business investment is clustered near educational hubs, so indirectly even if you didn't go to college and directly receive education your wage is function of the education of local peers. Is this an acceptable edge-case? Is it borne out by the majority distribution of colleges within urban centres? These are the kinds of questions we need be concerned with as we assess the relationship between our instrument and our outcome. \n",
     "\n",
@@ -1379,7 +1385,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### The Wald Estimate"
+    "#### The Wald Estimate"
    ]
   },
   {